Math 360 Notes
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Additional Maths Notes (20 Oct 2014)
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Table of Contents
Ex 1.1 Simultaneous Equations ................................................. 5
Solve a Pair of Linear & Non-linear Eqns .............................. 5
Form Relation ................................................................................... 5
Ex 1.2 Sum and Product of Roots .............................................. 5
Sum & Product of Roots ................................................................ 5
Form a Quadratic Equation from its Roots ........................... 5
Useful Formulae ............................................................................... 5
Prove Identities involving Roots ............................................... 5
Ex 1.3 Discriminant ........................................................................ 5
Complete the Square ...................................................................... 5
Sketch Quadratic graphs .............................................................. 6
Discriminant & Nature of Roots/Number of x-intercepts/Number of Intersections ....................................... 6
Conditions for ax2 + bx + c to be always positive or negative ............................................................................................... 6
Ex 1.4 Quadratic Inequalities ..................................................... 6
Solve Quadratic Inequality .......................................................... 6
Solve Simultaneous Inequalities ............................................... 6
Form Quadratic Inequality from Solution ............................. 6
Ex 2.1 Surds ...................................................................................... 6
Surds Properties .............................................................................. 6
Simplify Surds ................................................................................... 7
Rationalize Denominator ............................................................. 7
Solve Surds Equation ..................................................................... 7
Method of Difference ..................................................................... 7
Ex 2.2 Indices ................................................................................... 7
Law of Indices ................................................................................... 7
Ex 2.3 Index equations .................................................................. 7
Equality of Indices .......................................................................... 7
Different Types of Manipulation ............................................... 8
Ex 2.4 Exponential Functions ..................................................... 8
Sketch Exponential Functions .................................................... 8
Ex 3.1 Polynomials and Identities ............................................. 8
Definition of Polynomials ............................................................ 8
Multiply Polynomials .................................................................... 8
Find Unknown(s) in an Identity ............................................... 8
Ex 3.2 Division of Polynomials ................................................... 8
Long Division .................................................................................... 8
Division Algorithm ......................................................................... 9
Ex 3.3 Remainder Theorem ......................................................... 9
Remainder Theorem ..................................................................... 9
Ex 3.4 Factor Theorem .................................................................. 9
Factor Theorem ............................................................................... 9
Sum/Difference of Cubes ............................................................. 9
Ex 3.5 Cubic Polynomials and Equations ............................... 9
Factorize Cubic Expressions ...................................................... 9
Form Cubic Polynomial ................................................................ 9
Ex 3.6 Partial Fractions ................................................................. 9
Break into Partial Fractions ....................................................... 9
Cover-up Rule ................................................................................. 10
Compare Coefficients .................................................................. 10
Juggling ............................................................................................. 10
Proper & Improper fraction ..................................................... 10
Ex 4.1 Modulus Functions and their Graphs ...................... 10
Modulus Definition ...................................................................... 10
Modulus Properties ..................................................................... 10
Solve Modulus Equations .......................................................... 10
Sketch y = f(|x|) ............................................................................ 10
Ex 4.2 Power Graphs ................................................................... 11
Sketch Power Graphs .................................................................. 11
Ex 5.1 Binomial Expansion of ( + ) ............................... 11
Factorial ............................................................................................ 11
Combination ................................................................................... 11
Use Pascals Triangle ................................................................... 11
Expand (1 + b)n ............................................................................ 11
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Binomial Theorem Cross-applications ................................ 12
Ex 5.2 Binomial Expansion of ( + ) ................................12
Expand (a + b)n ............................................................................ 12
Use Tr+1 ............................................................................................ 12
Ex 6.1 Mid-point of a Line Segment .......................................12
Distance Formula ......................................................................... 12
Gradient ........................................................................................... 12
Find line ........................................................................................... 12
Use point on line/curve ............................................................. 13
Ratio of Diagonal Segments ..................................................... 13
Use Vectors ..................................................................................... 13
Find Intersection .......................................................................... 14
Mid-point Formula ...................................................................... 14
Ex 6.2 Parallel Lines .....................................................................14
Angle of Inclination ..................................................................... 14
Parallel Lines .................................................................................. 14
Collinearity ..................................................................................... 15
Find Parallel Line ......................................................................... 15
Ex 6.3 Perpendicular Lines .......................................................15
Perpendicular Lines .................................................................... 15
Find Perpendicular Line ............................................................ 15
Find Perpendicular Bisector .................................................... 15
Ex 6.4 Areas of Triangles and Quadrilaterals .....................15
Shoelace Formula ......................................................................... 15
Ex 7.1 Introduction to Logarithms .........................................15
Logarithm Definition .................................................................. 15
Special Log Values ........................................................................ 15
Convert between Log & Index Form .................................... 16
Ex 7.2 Laws of Logarithms ........................................................16
Laws of Logarithm ....................................................................... 16
Ex 7.3 Logarithmic Equations ..................................................16
Equality of Logarithms ............................................................... 16
Solve Log Equations .................................................................... 16
Ex 7.4 Log and Eqns of the form = ..............................16
Solve ax = b .................................................................................... 16
Solve Index Equations ................................................................ 16
Ex 7.5 Logarithmic Graphs ........................................................17
Draw Logarithmic Graphs ........................................................ 17
Ex 8.1 Reducing Equations to Linear Form ........................17
Linearize .......................................................................................... 17
Form Non-linear Equation ....................................................... 17
Equate Coordinates ...................................................................... 17
Ex 8.2 Linear Law ......................................................................... 17
Linearization ................................................................................... 17
Gradient & Y-intercept ............................................................... 17
Scale.................................................................................................... 17
Graphical Reading ........................................................................ 17
Intersection ..................................................................................... 18
Ex 9.1 Graphs of Parabolas of the Form = ............. 18
Sketch y2 = kx ............................................................................... 18
Ex 9.2 Coordinate Geometry of Circles ................................ 18
Circle Equation .............................................................................. 18
Circle Equation Cross-applications ....................................... 19
Ex 10.1 Triangle Theorems ............................................................ 20
Use Line Addition and Subtraction ........................................ 20
Angle Properties of Line(s) ...................................................... 20
Angle Properties of Triangles .................................................. 20
Congruency Tests ......................................................................... 20
Similarity Tests .............................................................................. 20
Mid-point Theorem ...................................................................... 21
Ex 10.2 Quadrilaterals Theorems ........................................ 21
Definition & Properties of Quadrilaterals ........................... 21
Prove Quadrilaterals ................................................................... 22
Ex 10.3 Circles Theorems ........................................................ 22
Angle Properties of Circle .......................................................... 22
Chord Properties of Circle ......................................................... 22
Tangent Properties of Circle .................................................... 22
Ex 11.1 Trigo Ratios of Acute Angles .................................. 22
Special Angles ................................................................................. 22
Convert between Degrees and Radians ............................... 23
Complementary s ....................................................................... 23
Supplementary s ........................................................................ 23
Identify Quadrant ......................................................................... 23
Find Basic Angle ........................................................................ 23
Find General Angle ................................................................... 23
Use .................................................................................................. 23
Ex 11.2 Trigo Ratios of any Angles ...................................... 23
Trigo Function Definition .......................................................... 23
Use in Quadrant(s) .................................................................. 23
Reciprocal Identities ................................................................... 24
Negative Angles ............................................................................. 24
ASTC Rule ......................................................................................... 24
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Solve Trigo Eqn f(x) = k ........................................................... 24
Ex 11.3 Trigo Graphs .................................................................25
Solve Trigo Eqn f(x) = k by Graph ........................................ 25
Range of Sine & Cosine ............................................................... 25
Find Unknowns of Trigo Function af(bx) + c ................... 25
Sketch Trigonometric Functions ........................................... 25
Use Symmetrical/Cyclical Nature of Trigo Graphs ........ 26
Inverse Trigo Function .............................................................. 27
Ex 12.1 Simple Identities .........................................................27
Questions involving Identities ................................................ 27
Ratio Identities .............................................................................. 27
Pythagorean Identities ............................................................... 28
Square Root of Trigo Function f(x) ....................................... 28
Ex 12.2 Further Trigo Eqns .....................................................28
Simplify to Tangent Eqn ............................................................ 28
Factorize Trigo Eqn ..................................................................... 28
Solve Trigo Eqn f(ax + b) = k ................................................. 28
Ex 13.1 The Addition Formulae ............................................28
Addition Formulae ....................................................................... 28
Ex 13.2 The Double Angle Formulae ...................................29
Double Formulae ...................................................................... 29
Half Formulae ............................................................................ 29
Ex 13.3 The R-Formulae ..........................................................29
R-Formulae ..................................................................................... 29
Ex 14.1 The Derivative and its Basic Rules .......................29
Derivative as Gradient ............................................................... 29
Power Rule ...................................................................................... 30
Constant Multiple Rule .............................................................. 30
Sum/Difference Rule .................................................................. 30
Differentiation from First Principles .................................... 30
Ex 14.2 The Chain Rule .............................................................30
Chain Rule ....................................................................................... 30
Ex 14.3 The Product Rule ........................................................30
Product Rule ................................................................................... 30
Ex 14.4 The Quotient Rule ......................................................31
Quotient Rule ................................................................................. 31
Ex 15.1 Tangents and Normals..............................................31
Find Tangent .................................................................................. 31
Find Normal .................................................................................... 31
Tangent Properties ...................................................................... 31
Normal Properties ....................................................................... 31
Ex 15.2 Increasing and Decreasing Functions ................ 31
Increasing/Decreasing function ............................................. 31
Ex 15.3 Rates of Change ........................................................... 31
Rate of Change ............................................................................... 31
Quantity & Constant Rate .......................................................... 31
Ex 15.4 Connected Rates of Change .................................... 32
Connected Rates of Change ...................................................... 32
Ex 16.1 Nature of Stationary Points .................................... 32
Stationary Point/Value ............................................................... 32
1st Derivative Test ....................................................................... 32
2nd Derivative Test ...................................................................... 32
Ex 16.2 Maxima and Minima .................................................. 32
Maxima/Minima ............................................................................ 32
Ex 17.1 Derivatives of Trigo Functions .............................. 32
Derivatives of Trigonometric Functions ............................. 32
Ex 17.2 Derivatives of Exponential Functions ................ 32
Derivatives of Exponential Functions .................................. 32
Ex 17.3 Derivatives of Log Functions ................................. 33
Derivatives of Log functions ..................................................... 33
Use Logarithmic Differentiation ............................................. 33
Ex 18.1 Indefinite Integrals .................................................... 34
Integral Rules ................................................................................. 34
Find Integral from Derivative .................................................. 34
Find Curve from Derivative ...................................................... 34
Integrals of Power Functions ................................................... 34
Ex 18.2 Definite Integrals ........................................................ 35
Definite Integrals .......................................................................... 35
Definite Integrals Rules .............................................................. 35
Integrals of Modulus Functions .............................................. 35
Ex 18.3 Integrals of Trigo Functions ................................... 35
Integrals of Trigonometric Functions .................................. 35
Ex 18.4 Integrals of Exponential Fns & 1/x ..................... 35
Integrals of Exponential Functions ....................................... 35
Integrals of 1/x & 1/(ax + b) ................................................... 35
Ex 19.1 Area by Integration ................................................... 36
Area by integration ...................................................................... 36
Ex 19.2 Area bounded by Curves ......................................... 36
Strategies to find area bounded by curves ......................... 36
Ex 20.1 Kinematics .................................................................... 36
Kinematics Relation ..................................................................... 36
Implications of Kinematics Statements ............................... 37
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Distance ............................................................................................ 37
Appendix 1 Geometric Formulae .............................................38
2D Shapes ........................................................................................ 38
3D Shapes ........................................................................................ 38
Appendix 2 Trigonometric Identities .....................................39
Appendix 3 Calculus Formulae .................................................40
Differentiation ............................................................................... 40
Integration ...................................................................................... 40
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Step 1: Subject variable in linear eqn Step 2: Substitute it into non-linear eqn
Step 1: Assign variables
Step 2: Form relation between variables
Step 1: Simplify to ax2 + bx + c = 0
Step 2: State roots
Step 3: Find SOR/POR
Sum of roots = + = b
a
Product of roots = =c
a
Applications Evaluate expressions involving its roots
e.g. find 2
+
2
Given context e.g. the heights of two men satisfy 40x2 138x + 119 = 0. Without solving the
equation, find the average height of these two
men. Average height =+
2
Find roots and unknowns e.g. the equation x2 4x + c = 0 has roots which
differ by 2. Find the value of each root and c.
To prove existence of positive & negative root, use
< 0
Convert to quadratic equation in y by substitution
e.g. x2
3 2x1
3 + 3 = 0 has roots &
sub y = x1
3:
y2 2y + 3 = 0 has roots 1
3 & 1
3
Step 1: State roots
Step 2: Find SOR/POR
Step 3: Form equation
x2 (SOR)x + (POR) = 0
2 + 2 = ( + )2 2
= ( )2
( )2 = ( + )2 4
4 + 4 = (2 + 2)2 2()2
To form useful equations (to be substituted),
(i) is a root of ax2 + bx + c = 0, a2 + b + c = 0 (1)
(ii) Multiply n to (1)
(iii) Apply power of n to (1)
Given that is a root of the equation x2 = x 3, show that (i) 3 + 2 + 3 = 0 (ii) 4 + 52 + 9 = 0
(i) is root, 2 = 3 (1)
(1) : 3 = 2 3 (2) sub (1) into (2): 3 = ( 3) 3 3 = 2 3 3 + 2 + 3 = 0 (shown)
(ii) (1)2: 4 = ( 3)2
4 = 2 6 + 9 4 = 2 6(2 + 3) + 9 [use (1) to make the subject] 4 = 2 62 18 + 9 4 = 52 + 9 4 + 5 9 = 0 (shown)
x2 + kx = (x +k
2)2 (
k
2)2
Solve a Pair of Linear & Non-linear Eqns
Ex 1.1 Simultaneous Equations
Form Relation
Sum & Product of Roots
Ex 1.2 Sum and Product of Roots
Form a Quadratic Equation from its Roots
Useful Formulae
Solution
Question
Prove Identities involving Roots
Complete the Square
Ex 1.3 Discriminant
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Step 1: Express as y = a(x h)2 + k
Step 2: Obtain turning point (h, k)
Step 3: Determine or shape from a
Step 4: Sub x = 0 to find y-intercept
Step 5: Sub y = 0 to find x-intercept Note: x = h is the line of symmetry e.g.
To find x1, 2+x1
2= 5 x1 = 8
Step 1: Simplify to ax2 + bx + c = 0
(by substituting line into curve)
Step 2: Use relation between b2 4ac & nature of roots/x-intercepts/intersections
Discriminant Nature of roots
No. of x-intercepts/ intersections
b2 4ac > 0 2 distinct 2 b2 4ac = 0 2 equal 1 (tangent) b2 4ac 0 2 1 or 2 (meet) b2 4ac < 0 0 0
ax2 + bx + c > 0 for all x a > 0, b2 4ac < 0
ax2 + bx + c < 0 for all x a < 0, b2 4ac < 0 Step 1: Simplify inequality to ax2 + bx + c vs 0
Step 2: Use 2 conditions (i) a > 0 or a < 0 (ii) b2 4ac < 0
Step 1: Simplify to ax2 + bx + c vs 0, a > 0 Step 2: Factorize Step 3: Draw sign diagram (Arrange roots & alternate signs with + at left)
Step 4: Find range of x
f(x) < g(x) < h(x) f(x) < g(x) and g(x) < h(x)
Step 1: Split into 2 inequalities using and
Step 2: Solve each inequality
Step 3: Take intersection of both solutions
x1 < x < x2 k(x x1)(x x2) < 0
x < x1 or x > x2 k(x x1)(x x2) > 0
For a > 0 and b > 0,
a b = ab
a
b =
a
b
a a = a Notation
For xn ,
n index
x radicand
radical sign or radix or root symbol
xn
surd (if xn is irrational)
Note: For xn and x < 0,
Even n results in non-real number
Odd n results in real number
e.g. 4 does not exist but 83
exists
2 (5, 3)
1
= 2 + +
Sketch Quadratic graphs
Discriminant & Nature of Roots/Number of x-intercepts/Number of Intersections
Conditions for ax2 + bx + c to be always positive or negative
1 2 + +
Solve Quadratic Inequality
Ex 1.4 Quadratic Inequalities
Solve Simultaneous Inequalities
Form Quadratic Inequality from Solution
Surds Properties
Ex 2.1 Surds
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Factor out largest square number
e.g. 45 = 9 5 = 35 Prime factorize (for more challenging numbers)
e.g. 540 = 22 33 5
= 2 31.5 5
= 2 33 5
= 615
1
a
a
a=
a
a
1
ah + bk
1
ah bk=
1
a2h b2k
Square both sides
a = b a = b2
Equate rational & irrational terms
a + bk = c + dk a = c, b = d
Note: Check the answer mentally by substituting it into
the original equation.
e.g. 6 5x = x 6 5x = x2 x2 + 5x 6 = 0 (x + 6)(x 1) = 0 x = 6 or x = 1 (rej) When x = 1,
LHS = 6 5 = 1 RHS = 1 LHS RHS
If you cannot simplify to a = b or a + bk = c + dk, consider solving surds equation by substitution
e.g. 2x + 3x + 1 = 0
sub u = x:
2u2 + 3u + 1 = 0
Step 1: Break each term into partial sums
Step 2: Arrange partial sums vertically
Step 3: Cancel diagonally
a0 = 1
an =1
an
am
n = (an )
m= am
n
(am)n = amn
When you multiply/divide terms, identify common base/power
e.g. 313309
13
2723
(common base is 3)
e.g. (a3 + b2 + b3
) (a3 + b2 b3
)
(common power is 1
3)
When you add/subtract terms, identify highest common factor e.g. 8x+2 34(23x)
= (23)x+2 2 17(23x)
= 23x+6 17(23x+1) (HCF is 23x+1)
= 23x+1(25 17)
= 23x+1(15)
Note: Equations involving even power functions may have multiple solutions
e.g. x4 = 16 x = 2 or x = 2
ax = an, for a > 0, a 1 x = n
Simplify Surds
Rationalize Denominator
Solve Surds Equation
Method of Difference
am an = am+n
am
an = amn
an bn = (ab)n
an
bn = (
a
b)n
Same Base
Same Power
Law of Indices
Ex 2.2 Indices
Equality of Indices
Ex 2.3 Index equations
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Manipulate Key Words Simplify Complex to simple Express In terms of Evaluate Find numerical value Show Work towards distinct characteristic Solve Equation Given Consider rearranging given equation.
y = ax, a > 1 (slopes up)
y = ax, 0 < a < 1 (slopes down)
Note: y = ax 0 for 0 < a < 1
Polynomials must have
non-negative power
integer power
Expand using strategic alignment
Find coefficient using selective multiplication
Substitute Compare coefficients Tip: Sub values of x that makes a factor zero e.g. a(x 2) + b = 5 3x
sub x = 2: a(0) + b = 5 3(2) b = 1
compare x: a = 3
Step 1: Surface out hidden terms and express polynomial
in powers of decreasing integers
Repeat step 2-5 until Deg(R) < Deg(divisor)
Step 2: Divide
Step 3: Multiply
Step 4: Subtract
Step 5: Bring down e.g. (3x2 2x + 5) (x + 2)
Different Types of Manipulation
1
1
Sketch Exponential Functions
Ex 2.4 Exponential Functions
Definition of Polynomials
Ex 3.1 Polynomials and Identities
e.g. Find coefficient of x2 in (x2 + x + 1)(x2 + 2x + 3)
e.g. (x + 1)(x2 + x + 1) = x3 +x2 +x +x2 +x +1
Multiply Polynomials
Find Unknown(s) in an Identity
Long Division
Ex 3.2 Division of Polynomials
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Quotient Q(x) Divisor g(x) Dividend f(x) Remainder R(x)
Dividend = Divisor Quotient + Remainder
Dividend
Divisor = Quotient +
Remainder
Divisor
Deg(Dividend) = Deg(Divisor) + Deg(Quotient)
Deg(Remainder) < Deg(Divisor)
For quadratic divisor, f(x) (px2 + qx + r) R(x) = ax + b f(x) = (px2 + qx + r)Q(x) + ax + b
Polynomial f(x) (ax + b) R = f (b
a)
Tip: Insert value of x that makes the divisor zero Note: If polynomial is not given, use division algorithm
Polynomial f(x) has factor (ax + b) f (b
a) = 0
Tip: Insert value of x that makes the factor zero
a3 b3 = (a b)(a2 ab + b2)
Step 1: Guess factor (factor thm)
Step 2: Compare x3
Step 3: Compare x0
Step 4: Compare x2
Step 5: Compare x (optional)
Given 1 distinct root x1,
f(x) = k(x x1)3
Given 2 distinct roots x1 and x2,
f(x) = k(x x1)2(x x2) or k(x x1)(x x2)
2 Given 3 distinct roots x1, x2 and x3,
f(x) = k(x x1)(x x2)(x x3)
Step 1: Convert to proper fraction
Step 2: Factorize denominator
Step 3: Break into partial fraction forms
Step 4: Solve for unknowns Cover-up rule Substitution Compare coefficients
Denominator Form
ax + b A
ax+b
(ax + b)2 A
ax+b+
B
(ax+b)2
x2 + c2 Ax+B
x2+c2
Division Algorithm
Remainder Theorem
Ex 3.3 Remainder Theorem
Factor Theorem
Ex 3.4 Factor Theorem
Sum/Difference of Cubes
(x )
(x )(px2 + + )
(x )(px2 + + r)
(x )(px2 + qx + r)
(x )(px2 + qx + r)
Factorize Cubic Expressions
Ex 3.5 Cubic Polynomials and Equations
Form Cubic Polynomial
Break into Partial Fractions
Ex 3.6 Partial Fractions
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To solve for unknown with linear factors,
Step 1: Insert root Step 2: Cover up linear factor (that becomes zero) Step 3: Equate unknown (highest power)
e.g. f(x)
(xx1)(xx2)2=
A
(xx1)+
B
(xx2)+
C
(xx2)2
insert x1, cover up (x x1) and equate A
x = x1: f(x)
( )(xx2)2|x=x1
= A
insert x2, cover up (x x2) and equate C
x = x2: f(x)
(xx1)( )2|x=x2
= C
Step 1: Clear fractions by multiplying denominator
Step 2: Simplify to polynomial of descending power
Step 3: Compare coefficients
e.g. x2+2x+15
x(x2+3) =
5
x+
Bx+C
x2+3
x2 + 2x + 15 = 5(x2 + 3) +(Bx + C)x
= 5x2 + 15 +Bx2 + Cx
= (5 + B)x2 + Cx + 15
Compare coefficients: x2: 1 = 5 + B B = 4 x: C = 2
x2+2x+15
x(x2+3)=
5
x+
4x+2
x2+3
Step 1: Copy denominator to numerator Step 2: Multiply to match term with highest power Step 3: Add to balance
e.g.
4x2 + 3
x2 2
(x2 2)
x2 2 copy (2 2) to the to numerator
4(x2 2)
x2 2
Multiply 4 to match term with highest power
=4(x2 2) + 11
x2 2 add 11 to numerator to balance
= 4 +11
x2 2
Divide each term in numerator by denominator
Proper Fraction: Deg(Numerator) < Deg(Denominator) Improper Fraction: Deg(Numerator) Deg(Denominator)
|x| = {x x 0
x x < 0
Tip: Use given condition to determine if |f(x)| = f(x) or f(x)
e.g. Given a > 2, simplify |3 2a|
a > 2
2a < 4
3 2a < 1
< 0
|3 2a| = (3 2a) = 2a 3
|a| = |a|
|ab| = |a||b|
|a
b| =
|a|
|b|
|an| = |a|n
|a|2 = |a2| = a2
Tip: differ is the trigger word to use modulus
e.g. A differs from B by 10 |A B| = 10
|f(x)| = g(x) f(x) = g(x) or f(x) = g(x) Note: Check the answer by mentally substituting it into the original equation. Tip: Consider squaring both sides to remove mod |f(x)| = k
[f(x)]2 = k2
Step 1: Sketch y = f(x)
Step 2: Reflect negative part of f(x) in x-axis
Cover-up Rule
Compare Coefficients
Juggling
Proper & Improper fraction
Modulus Definition
Ex 4.1 Modulus Functions and their Graphs
Modulus Properties
Solve Modulus Equations
Sketch y = |f(x)|
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For y = axn, a > 0
Integer
Integer Negative Positive
Even Odd Even Odd ( 1)
y =1
x2 y =
1
x y = x2 y = x3
Rational
Rational n < 0
( at rate)
0 < n < 1
( at rate)
n > 1
( at rate)
n! = n (n 1) 2 1
= n (n 1)!
0! = 1
To divide between factorials,
Step 1: Expand bigger factorial till smaller factorial
Step 2: Strike out smaller factorials
e.g. 7!
5!=
765!
5!= 7 6 = 42
(n+1)!
(n2)!=
(n+1)n(n1)(n2)!
(n2)!= (n + 1)n(n 1)
(nr) =
n!
(nr)!r!
(n0) = 1
(n1) = n
(n2) =
n(n1)
2
(n3) =
n(n1)(n2)
3!
(n4) =
n(n1)(n2)(n3)
4!
To create table,
n Binomial coefficients 0 1 1 1 1 2 1 2 1
Step 1: Insert 1 at the sides
Step 2: Form numbers inside triangle by adding the 2 numbers above it
Step 3: Use formula
(1 + b)n = (1st coeff)b0 +(2nd coeff)b1 ++ (last coeff)bn
(1 + b)n = (n0) b0 +(
n1) b1 ++ (
nr) br ++ (
nn) bn
= 1 +nb ++ (nr) br ++ bn
Sketch Power Graphs
Ex 4.2 Power Graphs
Factorial
Ex 5.1 Binomial Expansion of (1 + b)n
Combination
Use Pascals Triangle
Expand (1 + b)n
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App 1: Multiply selectively
App 2: Substitute value/terms
App 3: Compare coefficients
(i) Expand (2 x) (1 +1
2x)
8 in ascending powers of x, as
far as the term in x3.
(ii) Hence estimate the value of 1.9 (1.05)8
Multiply selectively
(i) (2 x) (1 +1
2x)
8
= 2 +8x +14x2 +14x3
x 4x2 7x3 +
= 2 +7x +10x2 +7x3 + Substitute values (ii) 1.9 (1.05)8 = (2 0.1) (1 + 0.5)8
= [2 (0.1)] [1 +1
2(0.1)]
8
= 2 +7(0.1) +10(0.1)2 +7(0.1)3 + [sub x = 0.1 into (i)]
2.807
The first three terms in the expansion, in ascending powers of x of (1 + 2x)n are 1 + 16x + ax2. Find n and a.
Compare coefficients (1 + 2x)n = 1 + 2nx + 2n(n 1)x2 + (by Binomial Thm) 1 + 16x + ax2 (given) Compare x: 2n = 16 n = 8
Compare x2 2n(n 1) = a Sub n = 8: 2(8)(8 1) = a a = 112
(a + b)n = (n0) an0b0 +(
n1) an1b1 ++ (
nn) annbn
= 1 +nan1b ++ bn
Tr+1 = (nr) anrbr
To find particular term,
Step 1: Simplify to (a + b)n
Step 2: Use Tr+1 Pull out x
Step 3: Find r Equate power
middle term r =n
2
constant power = 0
Step 4: Insert r into Tr+1
(x1 x2)2 + (y1 y2)2
m =y1 y2x1 x2
If y-intercept is not given,
Step 1: Find point
Step 2: Find gradient
Step 3: Find line y y1 = m(x x1)
If y-intercept is given,
Step 1: State y-intercept
Step 2: Find gradient
Step 3: Find line y = mx + c
Note: For m = 0, horizontal line y = c For m , vertical line x = a
Solution
Question
= (2 x)(1 +4x + 7x2 + 7x3 + ) [by Binomial Thm]
Solution
Question
Binomial Theorem Cross-applications
Expand (a + b)n
Ex 5.2 Binomial Expansion of (a + b)n
Use Tr+1
Distance Formula
Ex 6.1 Mid-point of a Line Segment
Gradient
Find line
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(x1, y1) lies on y = f(x)
y1 = f(x1) (form eqn)
(x1, f(x1)) (express coordinates in only 1 variable)
e.g. (2,1) lies on y = kx 2 1 = k(2) 2
2k = 3
k =3
2
e.g. A(a1, a2) lies on y = 2x
A(a1, 2a1)
By similar triangles, A: B (diagonal) = C:D (horizontal) = E: F (vertical)
AB = AO + OB
= OB OA
Given A is (0,9) & C is (6,3) and AB: BC = 2: 1, find the coordinates of B.
OB = OA +AB
= OA +2
2+1AC
= OA +2
3(OC OA )
= OA +2
3OC
2
3OA
=1
3OA +
2
3OC
=1
3(09) +
2
3(63)
= (45)
B(4,5)
A is (0,6), B is (2, 2) and D is (2, 2). AB is parallel to DC and AB: DC = 1: 2. Find the coordinates of C
OC = OD +DC
= OD +2AB
= OD +2(OB OA )
= OD +2 [(88) (
06)]
= OD +2(82)
= (2
2) +(
164
)
= (182
)
C is (18,2)
Use point on line/curve
Ratio of Diagonal Segments
Solution
Question
Solution
Question
Use Vectors
A
B
C E
F
D
(6,3)
(0,9)
2
1
(0,6)
(2,2)
(8,8)
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A is (2,4) and B is (6,10). ACB and MDB are 90. AC:MD = 3: 1. Find the coordinates of M
ACB ~ MDB. AB
MB =
AC
MD (corr. sides or ~ s)
=3
1
= 3
AB = 3MB
AM:MB = 2: 1
OM = OA +AM
= OA +2
3AB
= OA +2
3(OB OA )
=1
3OA +
2
3OB
=1
3(33) +
2
3(69)
= (57)
M is (5,7)
Solve a pair of equations
M = (x1 + x2
2,y1 + y2
2)
To find endpoint,
Step 1: Denote endpoint
Step 2: M = (x1+x2
2,y1+y2
2)
Step 3: Equate coordinates To find curve traced by mid-point,
Step 1: Find midpoint M
Step 2: Let M = (x, y)
Step 3: Equate coordinates
Step 4: Connect x & y
Shapes Implications Parallelogram ABCD
MAC = MBD
iso. ABC with AB = AC
MBC = Foot of from A to BC
Circle with diameter AB
MAB = Centre
A is (2,4) and B is (6,10). AC:MD = 2: 1. Given the diagram below, find the coordinates of M.
M = MAB = (2+6
2,4+10
2) = (4,7)
m = tan (wrt positive x axis)
l1 l2 m1 = m2
Solution
Question
Find Intersection
A B
C D M
A
B C
A B
Solution
2
1
(2,4)
(6,10)
Question
Mid-point Formula
Angle of Inclination
Ex 6.2 Parallel Lines
Parallel Lines
3
1
(3,3)
(6,9)
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If A, B & C are collinear, mAB = mAC (any two line segments)
Step 1: Find point/y-intercept
Step 2: Find gradient (m1 = m2)
Step 3: Find line y y1 = m(x x1)
y = mx + c
l1 l2 m1. m2 = 1
m1 =1
m2
Shapes Implications Rhombus ABCD
mAC mBD
P is equidistant from A and B
AB intersects P
Step 1: Find point/y-intercept
Step 2: Find gradient (m1 =1
m2)
Step 3: Find line y y1 = m(x x1)
y = mx + c
Step 1: Find mid-point (MAB)
Step 2: Find gradient (1
mAB)
Step 3: Find bisector (AB)
y y1 =1
mAB(x x1)
Tip: To find the line equidistant to points A & B, find the perpendicular bisector of AB If the 2 points have the same x or y-coordinate, bisector = average of the other coordinates
Area of polygon
=1
2|x1 x2 xn x1y1 y2 yn y1
|
=1
2[(sum of products )
(sum of product )] Coordinates should be in anti-clockwise order to have
positive output. On the contrary, if coordinates are in
clockwise order, the output is negative.
Use modulus if unsure anti-clockwise or clockwise
Zero area implies points are collinear
App 1: To find angle, use area =1
2ab sin C
App 2: To find height ( distance from point to line),
use Area =1
2(base)(height)
A logarithm must have
(i) base > 0
(ii) base 1
(iii) arg > 0
loga a = 1 Same base & argument results in output of 1
loga 1 = 0 argument of 1 results in output of 0
Collinearity
Find Parallel Line
A B
C D
P A
B
Perpendicular Lines
Ex 6.3 Perpendicular Lines
Find Perpendicular Line
Find Perpendicular Bisector
Area of quadrilateral 1
2|x1 x2 x3 x4 x1y1 y2 y3 y4 y1
|
=1
2(x1y2 + x2y3 + x3y4 + x4y1
x2y1 x3y2 x4y3 x1y4)
Area of triangle
=1
2|x1 x2 x3 x1y1 y2 y3 y1
|
=1
2(x1y2 + x2y3 + x3y1
x2y1 x3y2 x1y3)
Shoelace Formula
Ex 6.4 Areas of Triangles and Quadrilaterals
Logarithm Definition
Ex 7.1 Introduction to Logarithms
Special Log Values
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x = loga y y = a
x
Step 1: Identify base
Step 2: Connect base to opp. side of eqn
Step 3: Switch form keeping base
e.g. Log Index
Index Log
loga x = loga n x = n
Step 1: Use laws of log
Step 2: Remove log
Equality of log Change to index form
Step 3: Check log conditions
Note: Use substitution u = loga x if you cannot simplify to log =
To solve ax = b, log both sides
Method 1: ax = an x = n
Method 2: ax = b (log both sides)
Method 3: Convert to log form
Method 4: Substitution Step 1: Use laws of indices to simplify to ax = an or b
When multiply/divide terms, identify common base/power
When add/subtract terms/identify highest common factor
Step 2: Remove base
Equality of indices (ax = an x = n) Log both sides (ax = b) Convert to log form
If you cannot simplify to ax = an or b, use substitution u = ax
e.g. 9(3x)2 + 1 = 10(3x)
e.g. x3
2 8x3
2 = 7
ax = y x = loga y
Base is a. Connect base a to y
Switch from index to log form. Keep the base a, therefore argument is x.
loga y = x y = ax
Base is a. Connect base a to x
Switch from log to index form. Keep the base a, therefore power is x.
Convert between Log & Index Form
Product Law loga xy = loga x + loga y
Quotient Law logax
y = loga x loga y
Power Law loga xr = r loga x
Change-of-Base Law
loga b =logc b
logc a =
1
logb a
Laws of Logarithm
Ex 7.2 Laws of Logarithms
Equality of Logarithms
Ex 7.3 Logarithmic Equations
Laws of log Action
Change-of-base law convert to common base
Power law move coefficient to power
Product law/ Quotient law
combine to single log
Solve Log Equations
Solve ax = b
Ex 7.4 Log and Eqns of the form ax = b
Solve Index Equations
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y = loga x, a > 1 (slopes up)
y = loga x, 0 < a < 1 (slopes down)
Note: For base > 1, there is an inverse relation between base & rate of increase For a > b > c > 1,
Contains x & y
Contains constants
X & Y m & c
e.g. if ax2 + by3 = 1, then y3 = a
bx2 +
1
b
i.e. Y = y3, X = x2, m = a
b, c =
1
b
e.g. if y = eb+x
a , then ln y =1
ax +
b
a
i.e. Y = ln y , X = x,m =1
a, c =
b
a
To find unknowns,
Step 1: Linearize to axes variables
Step 2: Equate gradient & Y-intercept or use points on line
(whichever is given)
Step 1: Find point/gradient/ Y-intercept
Step 2: Form linear equation Y = mX + c (If Y-intercept is given) Y Y1 = m(X X1) (If Y-intercept is not given)
Step 3: Form non-linear eqn by replacing X & Y with axes variables
The first and second coordinates are not necessarily x and y respectively!
If C(9, 8) lies on the graph of yx against x, find the value of y corresponding to c.
Equate 1st coordinate: x = 9 Equate 2nd coordinate: yx = 8
y9 = 8
y = 8
3
Step 1: Simplify to Y = mX + c Step 2: Complete table
Step 1: State 2 points:
(i) On y-axis (ii) Halfway-down
Step 2: Equate gradient & Y-intercept
Step 1: Estimate Y-intercept
Y1 = mX1 + c c = Y1 mX1
Step 2: State domain & range Step 3: Find X & Y interval
X-Interval =XlastX1st
10
Y-Interval =YlastY1st
12
(Round down to 1, 2, 25 or 5) Step 4: State X & Y scale
Step 1: Simplify to (X or Y)
Step 2: Identify point
Step 3: Equate (Y or X) & solve for desired variable Note: Graphical reading is reliable only within the data range (interpolation) & not reliable outside the data range (extrapolation)
1
1
= log = log
= log
1
Draw Logarithmic Graphs
Ex 7.5 Logarithmic Graphs
Linearize
Ex 8.1 Reducing Equations to Linear Form
Form Non-linear Equation
Solution
Question
Equate Coordinates
Linearization
Ex 8.2 Linear Law
Gradient & Y-intercept
Scale
Graphical Reading
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Step 1: Work towards 2 curves on each side
Step 2: Plot 2nd curve & use intersection
y2 = kx, k > 0
Note: y = 0 is the line of symmetry
Given the graph y2 = 2x, draw a suitable line to solve x2 8x + 9 = 0.
x2 8x + 9 = 0 x2 6x + 9 = 2x (x 3)2 = 2x y = x 3 or y = (x 3)
Standard form (x a)2 + (y b)2 = r2
General form x2 + y2 + 2gx + 2fy + c = 0
Centre (a, b) = (g,f)
Radius r = g2 + f2 c
Note: It appears to be a counter-intuitive convention that g comes before f in the formula
Trigger/Setup Action
2 points Find bisector of chord where centre lies on
Centre & point Use distance formula to find radius.
Diameter Use midpoint formula
Touches horizontal/vertical line
Sketch graph. Deduce coordinates, centre, radius or point on circle. (see example)
Right angle triangle drawn
Use Pythagoras Theorem
0,1 or 2 intersections
Use discriminant.
Touches another circle
Connect centres with a line.
Line is tangent to circle
Identify right angle (tan rad) Find normal.
If the centre cannot be found from the approaches above (or only 1 coordinate can be deduced), use given information about centre (if any)
e.g. centre C(h, k) lies on line y = f(x) C is (h, f(h))
e.g. centre C(h, k) is 6 units away from point A(1,2)
(h 1)2 + (k 2)2 = 6 insert parameters into (x h)2 + (y k)2 = r2 and
solve for unknowns by elimination.
Intersection
Solution
Question
Sketch y2 = kx
Ex 9.1 Graphs of Parabolas of the Form y2 = kx
Circle Equation
Ex 9.2 Coordinate Geometry of Circles
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Examples of sketching graph to deduce information
Touch axis
Given: centre (3, 2), touches x-axis
Deduce: radius = 2
Cut axis
Given: Cuts y-axis at 2 and 5
Deduce: y coordinate of centre
=1+(5)
2= 3
Touch line(s)
Given: Touches x = 2 & x = 8
Deduce: radius =82
2= 3
x coordinate of centre
=2+8
2= 5
Pythagoras Theorem
Find length of PT, given radius is 13.
Idea: Find PC by distance formula
PT = PC2 CT2 (Pythagoras thm) Find AC
Idea: AC = r2 AB2 (Pythagoras thm)
Solve System of Equations
To find circle equation given 3 points on the circle,
insert the points into general form of circle
Complete the Square
Convert general form to standard form
Use Discriminant Find number of intersections between line & circle
(you can also compare the perpendicular distance
with the radius to determine the number of points of
intersection)
Find unknown c in line eqn given line is tangent to
circle
Find Intersection Point
Find point on circle
Find point of contact between tangent & normal
Find centre where line through centre meets
perpendicular bisector of chord
Use Distance Formula Find radius
To check if point A lies within circle, compare distance between A and centre with the radius
Use Midpoint Formula
Given that A(2,3) and B(4,5) are points on the circle,
the find the centre.
Given A is (2,3), the centre C is (4,5) and AB is the
diameter of the circle, find the point B
Find line Find tangent/normal at point of contact
e.g. Find AB
Find bisector
Whenever two points on circle are given, consider
finding the perpendicular bisector. The perpendicular
bisector of the chord passes through the centre of the
circle
Use Properties of Circle (refer to Ex 10.3)
bisector of chord
tan rad
(3, 2)
2
(1, 3)
1
5
= 2 = 8
3
(5, 2)
A C 2
6 r = 5
B
P(9,2)
T
C(2, 1)
Circle Equation Cross-applications
A B
P(3,10) 210
C(1, 4) AB
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AB + BC = AC
AC AB = BC
Given AB = CD, prove AC = BD
AB = CD AB + (BC) = CD + (BC) AC = BD
s in line opp. int. corr. alt. Prove straight lines by s in line = 180 Prove parallel lines by int., corr. & alt.
s in = 180
ext. = sum of
int. opp. s
iso. eq.
Prove equal sides/angles using iso. & eq.
SSS SAS AAS RHS 3 eq. sides
2 eq. sides, 1 included
2 eq. s, 1 corr. sides
1 rt , 1 eq. hyp, 1 eq. side
Note: Order of Points matter e.g. ABC XYZ is not the same as
ACB XYZ Prove equal sides/angles using congruent s
SSS SAS AA
3 sides 2 sides, 1 included
2 eq.
Given ABC ~ DEF, prove that AB DF = AC DE
Given ABC ~ DEF, prove that AB DF = AC DE
Whenever you encounter product of multiple line segments, consider using the property of similar triangles: ratios of corresponding sides are equal.
AB
AC
Identify which line segments in the above product correspond to the triangle ABC. AB and AC.
Take ratio AB
AC at the left. Note the
sequence. AB is 12 and AC is 13. 12 over 13.
AB
AC =
DE
DF
Use same sequence on the other triangle DEF at the right. 12 is
DE and 13 is DF. Take ratio DE
DF at
the right.
AB DF = AC DE [proven]
Cross multiply.
Given ABC ~ DEF & DE: EF = 1: 2, prove that
AB =1
2BC (or AB: BC = 1: 2)
AB
BC =
DE
EF
=1
2
AB =1
2BC
AB: BC = 1: 2
Solution
A B C D
Question
A B C
Use Line Addition and Subtraction
Ex 10.1 Triangle Theorems
a b a b
a b a
b a b
Angle Properties of Line(s)
a b c a b c a b a b a b
c
Angle Properties of Triangles
Congruency Tests
Solution
Question (Prove relation/ratio of line segments)
Solution
Question (prove product of sides)
Similarity Tests
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Given ABC ~ EDC, BC: CD = 1: 2 & area of ABC = x, find the area of DEC
Area of DEC = (2
1)2x = 4x [use
A1
A2= (
l1
l2)2]
D = MAB, E = MAC
DE BC, DE =1
2BC
Kite
Quad. with two pairs of equal adjacent sides
s between unequal sides are equal (angle)
One diagonal bisects the other (diagonal)
Longer diagonal bisects s (diagonal)
Diagonals are (diagonal)
Note: Concave kite have interior s > 180 Trapezium
Quad. with exactly one pair of parallel sides
supplementary interior s Parallelogram
Quad. with two pairs of parallel sides
Opp. sides are equal (side)
Opp. s are equal (angle)
interior s are supplementary (angle)
Diagonals bisect each other (diagonal) Rectangle
Quad. with four right angles Opp. sides are parallel (side)
Opp. sides are equal (side)
Diagonals bisect each other (diagonal)
Diagonals are equal (diagonal) Rhombus
Quad. with four equal sides
Opp. sides are parallel (side)
Supplementary interior s (angle)
Diagonals bisect s (diagonal)
Diagonals are bisector of each other (diagonal) Square
Quad. with four equal sides & four right angles
Diagonals bisect angles (diagonal)
Diagonals are equal (diagonal)
Diagonals are bisector of each other (diagonal)
Solution
Question (Use ratio of area of similar triangles)
A D
B C
E
Mid-point Theorem
Definition & Properties of Quadrilaterals
Ex 10.2 Quadrilaterals Theorems
A
B C
D
E 1
2
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Parallelogram
2 pairs of sides (definition)
2 pairs of equal & opp. sides (side)
1 pair of equal & sides (side)
2 pairs of equal opp. s (angle)
Diagonals bisect each other (diagonal) Rectangle
4 right s (definition)
Parallelogram + 1 right (angle) Rhombus
4 equal sides (definition)
Parallelogram + eq. adj. sides (side)
Parallelogram + bisecting diagonals (diagonal)
Parallelogram + diagonals (diagonal) Square
4 equal sides & 4 right s (definition)
Rectangle + eq. adj sides (side)
Rhombus + 1 right (angle) Trapezium
Parallel opposite sides (definition) Kite
2 pairs of equal adjacent sides (definition)
in
semicircle at centre = 2 at
circumference
s in same segment
s in opp. segment
bisector of chord passes through centre
Equal chords are equidistant from
centre Equal arcs results in equal chords
alt. segment
thm tan rad tangents
from ext. point
Table
0 30 45 60 90
0
6
4
3
2
sin 0 1
2 2
2
3
2 1
cos 1 32
2
2
1
2 0
tan 0 1
3 1 3
Triangle
Unit circle
Prove Quadrilaterals
a
a
b a b a
b
Angle Properties of Circle
Ex 10.3 Circles Theorems
A C B
O B
A X
Y C O
D
Chord Properties of Circle
a b O P
Q
R
Tangent Properties of Circle
45 60
30
60
1
1 2
2
1
3
Special Angles
Ex 11.1 Trigo Ratios of Acute Angles
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rad = 180
To convert from degrees to radians, multiply
180
To convert from radians to degrees, multiply 180
Tip:
Track the unit conversion to avoid the mistake of
multiplying the wrong fraction
e.g.
60 = 60
180 =
1
3
[deg] = [deg] [rad]
[deg] = [rad]
60 = 60 180
=
10800
[deg] = [deg] [deg]
[rad] =
[deg]2
[rad]
sin(90 ) = cos
cos(90 ) = sin
tan(90 ) =1
tan
sin(180 ) = sin
cos(180 ) = cos
tan(180 ) = tan
Step 1: Add or subtract 360 until 0 360 Step 2: Use table
Angle Quadrant 0 < < 90 1
90 < < 180 2 180 < < 270 3 270 < < 360 4
Step 1: Add or subtract 360 until 0 360
Step 2: Use table
Quadrant 1 2 180 3 180 4 360
Quadrant 1 2 180 3 180 + 4 360
Step 1: Draw
Step 2: Find all 3 sides (by Pythagoras Thm)
Angles measured anti-clockwise from the positive x-axis
are positive.
On the contrary, angles measured clockwise from the
positive x-axis are negative.
Step 1: Identify quadrant
Step 2: Draw in quadrant
Step 3: Find coordinates
Convert between Degrees and Radians
Complementary s
Supplementary s
Identify Quadrant
Find Basic Angle
Find General Angle
Use
x
y r
sin =y
r
cos =x
r
tan =y
x
r = x2 + y2
Trigo Function Definition
Ex 11.2 Trigo Ratios of any Angles
Use in Quadrant(s)
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Additional Math Notes (20 Oct 2014)
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Given that tan A =
5
12 and that tan A and cos A have
opposite signs, find the value of each of the following.
(i) sin(A)
(ii) cos(A)
(iii) tan (
2 A)
Thought Process
Step 1: Identify quadrants
tan A = 5
12< 0
2nd or 4th quad.
Observe that ratio for tan is negative. Tan is only positive in 1st or 3rd quad. Therefore, it is in 2nd or 4th quad.
tan A & cos A have opp. signs 3rd or 4th quad.
In 3rd quad., only tan is positive In 4th quad., only cos is positive Therefore, it is in 3rd or 4th quad.
4th quadrant Take overlap of above deductions. Therefore it is in 4th quadrant.
Step 2: Draw in quadrant
Draw in 4th quadrant.
Step 3: Find coordinates
tan A = 5
12=
y
x
tan A =y
x by definition.
y = 5,
Equate numerator, = 5. y-coordinate is negative in 4th quad.
x = 12, Equate denominator, = 12. x-coordinate is positive in 4th quad.
r = 122 + (5)2
= 13
Find hypotenuse r by Pythagoras Theorem
sin A =y
r=
5
13,
cos A =x
r=
12
13
Find other trigo ratios to serve as useful inputs. The rest of the question makes use of the 3 basic trigo ratios: sin A , cos A & tan A.
sec =1
cos
csc =1
sin
cot =1
tan
cos() = cos()
sin() = sin()
tan() = tan()
All trigo functions can be converted to trigo function of basic angle with positive or negative sign depending on ASTC rule. e.g. sin(210) = sin(30)
Quadrants method
Step 1: Find = f1(|k|) & identify quadrants
Step 2: State interval
Step 3: Find x using quadrants
5
12
r
Solution
Question Reciprocal Identities
Negative Angles
A S T C
sin is + all are +
tan is + cos is +
ASTC Rule
1 2
2 3 4 +
1 2
360 3 4 180 180 +
Solve Trigo Eqn f(x) = k
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Graphical method
When = 0 or 90,
i.e. sin f(x) = 0,1
cos f(x) = 0,1
tan f(x) = 0
Step 1: State interval
Step 2: Find x using graph
y = sin x y = cos x y = tan x
1 sin x 1
1 cos x 1
y = sin x y = cos x
Min sin x = 1
at x = 270 cos x = 1 at x = 180
Max sin x = 1 at x = 90
cos x = 1 at x = 0, 360
Sine/Cosine Tangent
Amplitude A = |a| =maxmin
2
Period T =360
b
Axis c =max+min
2
= min + A = max A
Period T =180
b
Step 1: Simplify to y = af(bx) + c Step 2: Find amplitude & period
Sin/Cos Tan Amplitude |a| Nil
Period 2
b
b
Step 3: Complete table and sketch graph
Domain x1 x x2 Axis with Amplitude
y = c |a|
Shape sin/cos/tan
Cycle x2x1
T
y = sin x y = cos x y = tan x
0 sin x = 0 at x =0, 180, 360
cos x = 0 at x =90, 270
tan x = 0 at x =0, 180, 360
Min sin x = 1 at x = 270
cos x = 1 at x = 180
Nil
Max sin x = 1 at x = 90
cos x = 1 at x = 0, 360
Nil
1
1
180 90 270
360 1
1
180 90 270
360 1
1
180 90 270
360
Solve Trigo Eqn f(x) = k by Graph
Ex 11.3 Trigo Graphs
1
1
180 90 270 360
1
1
180 90 270 360
Range of Sine & Cosine
c A = |a|
T =360
b
A = |a|
max
min
c
T =180
b
Find Unknowns of Trigo Function af(bx) + c
1
1
180 90 270 360
1
1
180 90 270 360
180 90 270 360
Sketch Trigonometric Functions
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Sketch y = 3(1 2 cos 4x) for 0 x 270
Step 1: Simplify to = () + y = 3(1 2 cos 4x)
= 3 6 cos 4x = 6 cos 4x + 3
Step 2: Find amplitude & period A = |6| = 6
T =360
4= 90
Step 3: Complete table and sketch graph
Domain 0 x 270 Axis with Amplitude
y = 3 6
Shape cos
Cycle 2700
90= 3
Follow the sequence from top down to sketch the graph.
Mark the endpoint of domain, 270.
Mark the axis 3. Add and subtract 6 to get max 9 and min 3.
Draw 1 cycle of negative cosine.
There are 3 cycles in total. Draw 2 more.
Symmetrical
Given & are roots of 3 cos x + 2 = 2 where 3 < k < 4. Find in terms of , given that <
x = is line of symmetry, +
2 =
= 2 Cyclical
Given that is the smallest positive root of the equation
2 cos 4x = 3.1 tan 2x, where 0 x 360, state the other roots in terms of .
Period = 90, x = , + 90, + 180, + 270
Solution
Question
270
270
3
9
3
270
3
9
3
270
3
9
3
1 = 3.1 tan 2
2 = 2 cos 4
90
= 45 = 135
180
2
2
Solution
Question
x
y
2
5
-1 2
= 3 cos + 2
Solution
Question
Use Symmetrical/Cyclical Nature of Trigo Graphs
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Principal values
2 sin1 x
2
0 cos1 x
2< tan1 x 0
For decreasing function, dy
dx< 0
Applications Determine whether a function is increasing or
decreasing Find the range of values of x for which a function is
increasing or decreasing
dy
dt rate of change of y wrt t
Consider adding/subtracting between related rates
Water is entering a container at a constant rate of 5 cm3/s Water is leaking from the container at a constant rate of 1 cm3/s. Find the net rate of water flow into the container.
Net rate = 5 1 = 4cm3/s
Quantity = (constant rate) time
d
dx[f(x)
g(x)] =
g(x) f(x) f(x) g(x)
[g(x)]2
Square Bottom
Diff Top
Bottom
Diff Bottom
Top
Quotient Rule
Ex 14.4 The Quotient Rule
Find Tangent
Ex 15.1 Tangents and Normals
Find Normal
Normal Properties
Tangent Properties
Increasing/Decreasing function
Ex 15.2 Increasing and Decreasing Functions
Solution
Question
Rate of Change
Ex 15.3 Rates of Change
Quantity & Constant Rate
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Step 1: Assign variables, state given & unknown rate
Step 2: y = f(x) (Form relation between variablesof given rate & unknown rate
)
(see appendix 1)
Step 3: dy
dx= f(x) (Find derivative)
Step 4: dy
dt=
dy
dxdx
dt (Use chain rule)
Step 5: Find rate at instant
dy
dx= 0
x a a a+ dy
dx sign + 0
max
x a a a+ dy
dx sign 0 +
min
x a a a+ dy
dx sign 0
inflexion
d2y
dx2< 0 max
d2y
dx2> 0 min
d2y
dx2= 0 inflexion
Step 1: Assign variables
Step 2: y = f(x) (Express variable to be max/minas a function of a single variable
)
(see appendix 1)
Step 3: Find dy
dx (Find derivative)
Step 4: Solve dy
dx= 0 (Find stationary value)
Step 5: Find d2A
dx2 and compare against 0
(Verify max/min)
d
dx(sin x) = cos x
d
dx(cos x) = sin x
d
dx(tan x) = sec2 x
d
dx[sin(ax + b)] = a cos(ax + b)
d
dx[cos(ax + b)] = a sin(ax + b)
d
dx[tan(ax + b)] = a sec2(ax + b)
Consider simplifying using trigonometric identities before differentiating
e.g. d
dx(2 sin x cos x) =
d
dx(sin 2x) = 2 cos x
d
dx(ex) = ex
d
dx(eax+b) = aeax+b
Consider simplifying using indices properties before differentiating
e.g. d
dx(e2x e13x) =
d
dx(e1x) = e1x
Connected Rates of Change
Ex 15.4 Connected Rates of Change
Stationary Point/Value
Ex 16.1 Nature of Stationary Points
1st Derivative Test
2nd Derivative Test
Maxima/Minima
Ex 16.2 Maxima and Minima
Derivatives of Trigonometric Functions
Ex 17.1 Derivatives of Trigo Functions
Derivatives of Exponential Functions
Ex 17.2 Derivatives of Exponential Functions
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d
dx(ln x) =
1
x
d
dx[ln(ax + b)]=
a
ax+b
Consider simplifying by using laws of logarithm before differentiating Product law
e.g. d
dx(ln xex) =
d
dx(ln x + ln ex)
=d
dx(ln x + 1)
=1
x
Quotient law
e.g. d
dx[ln (
x
x2+1)] =
d
dx[ln x + ln(x2 + 1)]
=1
x+
2x
x2+1
Power law
e.g. d
dx[ln(4x 3)2] =
d
dx[2 ln(4x 3)]
= 2(4
4x3)
=8
4x3
Change-of-base law
e.g. d
dx(loga x) =
d
dx(ln x
ln a)
=1
ln a
d
dx(ln x)
=1
ln a(1
x)
=1
xln a
Step 1: Take natural log both sides Step 2: Simplify using laws of log Step 3: Differentiate It is useful when differentiating
functions of the form y = [f(x)]g(x), f(x) e complicated products or quotients
Differentiate y = 2x with respect to x
y = 2x
ln y = ln 2x Take ln both sides
ln y = x ln 2 Simplify using power law
Diff wrt x: Differentiate both sides wrt x 1
ydy
dx = ln 2
dy
dx = (ln 2)y
= (ln 2)2x Replace y with 2x
Find dy
dx if y = (2 + x2)(1 x3)4
y = (2 + x2)3(1 x3)4 (1)
ln y = ln[(2 + x2)3(1 x3)4]
= ln(2 + x2)3 + ln(1 x3)4
= 3 ln(2 + x2) +4 ln(1 x3)
Diff wrt x: 1
ydy
dx = 3
1
2+x2
d
dx(2 + x2) +4
1
1x3
d
dx(1 x3)
= 3 1
2+x2 2x +4
1
1x3 (3x2)
=6x
2+x2
12x2
1x3
dy
dx = (
6x
2+x2
12x2
1x3) y
= 6x (1
2+x2
2x
1x3) y
= 6x [(1x3)2x(2+x2)
(2+x2)(1x3)] y
= 6x [1x34x2x3
(2+x2)(1x3)] y
= 6x [14x3x3
(2+x2)(1x3)] y (2)
sub (1) into (2): dy
dx = 6x [
14x3x3
(2+x2)(1x3)] (2 + x2)3(1 x3)4
= 6x(1 4x 3x3)(2 + x2)2(1 x3)3
Derivatives of Log functions
Ex 17.3 Derivatives of Log Functions
Solution
Question
Solution
Question
Use Logarithmic Differentiation
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f(x) g(x) dx = f(x) dx g(x) dx
af(x) dx = a f(x) dx
Integration is the reverse of differentiation
Given d
dx6x + 5 =
3
6x+5,
find 1
6x+5dx.
1
6x+5dx =
1
3
3
6x+5dx
=1
36x + 5 + c
Consider rearranging equation involving derivative.
Given d
dx(x ln x) = 1 + ln x,
find ln x dx.
d
dx(x ln x) = 1 + ln x
ln x =d
dx(x ln x) 1
ln x dx = [d
dx(x lnx) 1] dx
= x ln x x + c
Given d
dx(x cos x) = cos x x sin x,
Find x sin x dx
d
dx(x cos x) = cos x x sin x
x sin x = cos x d
dxx cos x
x sin x dx = (cos x d
dxx cos x) dx
= sin x x cos x + c
To form equations and solve unknowns, use given equations (unknowns are already present)
e.g. dy
dx= x2(x k)
use proportionality e.g. Gradient is proportional to f(x)
dy
dx= kf(x)
introduce arbitrary constants from integration
e.g. dy
dx= 2x + 1
y = x2 + x + c
use point on curve e.g. (1, 2) lies on y = f(x)
2 = f(1)
use gradient
e.g. at turning point, dy
dx= 0
xn dx =xn+1
n+1+ c
axn dx =axn+1
n+1 +c
(ax + b)n dx =(ax+b)n+1
a(n+1) +c
Note: The rules for above hold for all real values of n
except for n = 1
e.g. x1 dx x0
0+ c
but x1 dx = 1
xdx = ln|x| + c
Consider simplifying before integrating Multiply or divide
e.g. [x(x + 1)] dx = (x2 + x) dx =x3
3+
x2
2+ c
(2x2+4x
x) dx = (2x + 4) dx = x2 + 4x + c
(x2+2x
x1) dx = (x + 3 +
3
x1) dx (long division)
Use law of indices
e.g. x dx = x1
2 dx =x32
3
2
=2
3xx
1
x2dx = x2 dx =
x1
1+ c =
1
x+ c
Breaking into partial fractions
e.g. x
(x1)2dx =
1
x1+
1
(x1)2dx
= ln|x 1| 1
x1+ c
Integral Rules
Ex 18.1 Indefinite Integrals
Solution
Question
Solution
Question
Solution
Question
Find Integral from Derivative
Find Curve from Derivative
Integrals of Power Functions
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f(x)b
a dx = F(b) F(a)
f(x)b
a dx = f(x)
a
bdx
f(x)b
adx = a
cf(x) dx + f(x)
b
cdx
f(x)a
adx = 0
Definite integrals can be equal
because they have equal area
under curve
e.g. 01x2dx = x2
0
1dx
|f(x)| dx = { f(x) dx if f(x) 0
f(x) dx if f(x) < 0
sin x dx = cos x + c
cos x dx = sin x + c
sec2 x dx = tan x + c
sin(ax + b) dx = 1
acos(ax + b) + c
cos(ax + b) dx =1
asin(ax + b) + c
sec2(ax + b) dx =1
atan(ax + b) + c
Consider simplifying using trigonometric identities before integrating e.g. tan2 x dx = sec2 x 1dx
= tan x x + c Use special angles for definite integrals of trigonometric function
e.g. cos x
2
3
dx = [sin x]3
2
= sin
2 sin
3
= 1 3
2
ex dx = ex + c
eax+b dx =1
aeax+b + c
Consider simplifying using indices properties before integrating e.g. e2x e13x dx = e1x dx
=e1x
1 +c
= e1x +c
1
xdx = ln|x| + c
1
ax+bdx =
1
aln|ax + b| + c
Consider breaking into partial fractions before integrating
e.g. 2x1
(x+1)(x+2)dx = (
5
x+2
3
x+1) dx
= 5 ln|x + 2| 3 ln|x + 1| + c
Definite Integrals
Ex 18.2 Definite Integrals
Definite Integrals Rules
Integrals of Modulus Functions
Integrals of Trigonometric Functions
Ex 18.3 Integrals of Trigo Functions
Integrals of Exponential Functions
Ex 18.4 Integrals of Exponential Fns & 1/x
Integrals of 1
x &
1
ax+b
1 1
= 2
Equal areas
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Area = (Top Bottom)x2
x1
dx
Area = (Right Left)y2
y1
dy
There should be a pair of lines parallel to x or y-axis enclosing the region. If parallel to y-axis, integrate wrt x and vice versa. Consider finding geometric area without integration
Triangle area =1
2(base)(height)
Trapezium area =1
2(sum of bases)(height)
Given the the diagram at the right. Find area of (i) Region A (ii) Region B
(i) Area of Region A ()
=1
2(base)(height)
=1
2(1)(2)
= 1 unit2
(ii) Area of Region B (trapezium)
=1
2(sum of bases)(height)
=1
2(2 + 6)(3 1)
= 8 unit2
Axis Integrate wrt x or y-axis
Break Break into smaller shapes
Complement Subtract area
Find the area bounded by y = x2, y = 2 x and the x axis.
Method 1 (integrate wrt y-axis)
Area of region F
= (Right Left)y2y1
dy
= [(2 y) y]1
0dy
Method 2 (break)
Area of region G +Area of region H
= x21
0dx + (2 x)
2
1dx
Method 3 (complement)
Area of Area of region I
=1
2(2)(2) (2 x)
1
0dx
1 2
= ()
Top
Bottom
1
= ()
Left
2
Right
Solution
Question
Area by integration
Ex 19.1 Area by Integration
Question
= 2
= 2 2
(1,1)
Solution
F
=
= 2
1
G
= 2
= 2 1
H
2
= 2
= 2 2 1
I 2
Strategies to find area bounded by curves
Ex 19.2 Area bounded by Curves
v
v =ds
dt a =
dv
dt
v = a dt s = v dt
s a
Kinematics Relation
Ex 20.1 Kinematics
(1,2)
(3,6)
1 3
= 2
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t is time after passing O s|t=0 = 0
Rest v = 0
Time to turn around v = 0
Max/min quantity 1st derivative = 0
Max/min dist. from O v = 0
Max/Min v a = 0
Total Distance = |v| dt
Average Speed =total distance
total time
Total Distance = |v|t2
t1
dt
(Total distance travelled in between t1 and t2)
Method 1 (using s-t graph)
Step 1: Let v = 0 to find t
Step 2: Find s for each t found
Step 3: Find s for start & end
Step 4: Draw s-t graph Method 2 (using v-t graph)
Step 1: Draw v-t graph
Step 2: Use distance = |v| dt
Step 3: Split at v = 0
Step 4: Remove modulus
|v| = {v for v 0
v for v < 0
Implications of Kinematics Statements
Distance
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Triangle ( )
Triangle area =1
2(base)(height)
=1
2ab sin C
Isosceles triangle area =s23
4
Sine rule: sin a
A=
sinb
B=
sin c
C
Cosine rule: a2 = b2 + c2 2bc cos A
Pythagoras theorem : a2 + b2 = c2
Similar triangles: a
A=
b
B=
c
C
Trigonometric identities:
sin =o
h o = h sin
cos =a
h a = h cos
tan =o
a
Quadrilateral ( )
Square area = x2
Rectangle area = (base)(height)
Parallelogram area = (base)(height)
Rhombus area =1
2(product of diagonals)
Trapezium area =1
2(sum of bases)(height)
Kite area =1
2(product of diagonals)
Circle ( )
Circle area = r2
Circumference = 2r
Arc length = r = s
Area of sector =1
2r2
=1
2rs
Area of segment =1
2r2
1
2r2 sin
Circle properties (refer to Ex 10.3)
Prism
Prism volume = (base area)(height)
Cube volume = x3
Cube surface area = 6x2
Cylinder volume = r2h
Cylinder surface area = 2r2 + 2rh = 2r(r + h)
Pyramid
Pyramid volume =1
3(base area)(height)
Cone Volume =1
3r2h
Cone area (exclude base) = rl
where l = r2 + h2
Sphere
Sphere volume =4
3r3
Sphere area = 4r2
2D Shapes
Appendix 1 Geometric Formulae 3D Shapes
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Special Angles
0 30 45 60 90
0
6
4
3
2
sin 0 1
2 2
2
3
2 1
cos 1 32
2
2
1
2 0
tan 0 1
3 1 3
Complementary Angles
sin(90 ) = cos
cos(90 ) = sin
tan(90 ) =1
tan
Trigonometric Function Definition
Reciprocal Identities
sec =1
cos
csc =1
sin
cot =1
tan
Negative Angles
cos() = cos()
sin() = sin()
tan() = tan()
Principal Values
2 sin1 x
2
0 cos1 x
2< tan1 x