10.4 Solve Trigonometric Equations. Solving trig equations… What’s the difference?
Created by Mr. Lafferty Trigonometry Equations National 5 Solving Trig Equations Negative Cosine...

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Transcript of Created by Mr. Lafferty Trigonometry Equations National 5 Solving Trig Equations Negative Cosine...
created by Mr. Laffertycreated by Mr. Lafferty
Trigonometry EquationsTrigonometry Equationsw
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National 5
Solving Trig Equations
Negative Cosine
Trig Function and Circle Connection
Special Trig Relationships
Exam Type Questions
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StarterStarterw
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National 5
Q1. How can we tell if two lines are parallel.
Q2. Write down the three ratios connecting
the circle , arc length and area of a sector.
created by Mr. Laffertycreated by Mr. Lafferty
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Understand the Understand the connection between the connection between the circle and sine, cosine circle and sine, cosine and tan functions.and tan functions.
2. Solve trig equations using graphically.
1. We are investigating the connect between the circle and trig functions.
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Solving Trig Equations Solving Trig Equations National 5
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Trig and Circle Connection Trig and Circle Connection
All +veSin +ve
Tan +ve Cos +ve
180o  xo
180o + xo 360o  xo
1 2 3 4
National 5
Sine Graph Construction
Cosine Graph Construction
Tan Graph Construction Demo
created by Mr. Laffertycreated by Mr. Lafferty
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Use the balancing method Use the balancing method to trig equationto trig equation
a sin xo + 1 = 0
2. Realise that there are many solutions to trig equations depending on domain.
1. We are learning how to solve trig equations of the form
a sin xo + 1 = 0
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Solving Trig Equations Solving Trig Equations National 5
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Solving Trig EquationsSolving Trig Equationsa sin xo + b = 0
Example :
Solving the equation sin xo = 0.5 in the range 0o to 360o
Graphically what are we
trying to solve
xo = sin1(0.5)
xo = 30o
There is another solution
xo = 150o
(180o – 30o = 150o)
sin xo = (0.5)
1 2 3 4
National 5
Demo
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Solving Trig EquationsSolving Trig Equationsa cos xo + b = 0
Example :
Solving the equation cos xo = 0.625 in the range 0o to 360o
cos xo = 0.625
xo = 51.3o
(360o  53.1o = 308.7o)
xo = cos 1 0.625
There is another solution
1 2 3 4
National 5
Graphically what are we
trying to solveDemo
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Solving Trig EquationsSolving Trig Equationsa tan xo + b = 0
Example :
Solving the equation tan xo – 2 = 0 in the range 0o to 360o
tan xo = 2
xo = 63.4o
x = 180o + 63.4o = 243.4o
xo = tan 1(2)
There is another solution
1 2 3 4
National 5
Graphically what are we
trying to solveDemo
Example :
Solving the equation 3sin xo + 1 = 0 in the range 0o to 360o
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Solving Trig EquationsSolving Trig Equationsa sin xo + b = 0
sin xo = 1/3
Calculate first Quad valuexo = 19.5o
x = 180o + 19.5o = 199.5o
( 360o  19.5o = 340.5o)
There is another solution
1 2 3 4
National 5
Graphically what are we
trying to solveDemo
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Solving Trig EquationsSolving Trig Equationsa sin xo + b = 0
Example :
Solving the equation 2sin xo + 1 = 0 in the range 0o to 720o
sin xo = 1/2
Calculate first Quad valuexo = 30o
xo = 210o and 330o
360o + 210o = 570o
There are further solutions at
National 5
360o + 330o = 690o
Graphically what are we
trying to solveDemo
created by Mr. Laffertycreated by Mr. Lafferty
Now try N5 TJ Ex20.1 Q4 to Q10
Ch 20 (Page 198)
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Solving Trig EquationsSolving Trig Equations
National 5
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Outcome 3Higher
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Solving Trig Equations
Example
Solving the equation cos2x = 1 in the range 0o to 360o
Graphically what are we
trying to solve
cos xo = ± 1
cos xo = 1
cos2 xo = 1
xo = 0o and 360o
C
AS
T0o180
o
270o
90o
cos xo = 1xo = 180o
created by Mr. Laffertycreated by Mr. Lafferty
Now try N5 TJ Ex20.1 Q4 onwards
Ch 20 (Page 198)
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Solving Trig EquationsSolving Trig Equations
National 5
19 Apr 202319 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.
Starter QuestionsStarter Questionsw
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2
1. I f lines have the same gradient
What is special about them.
2. Factorise x +4x 12
3. Find the missing angles.
Nat 5
54o
19 Apr 202319 Apr 2023 Created by Mr. Lafferty Maths Dept.Created by Mr. Lafferty Maths Dept.ww
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1.1. Know what a negative Know what a negative cosine ratio means.cosine ratio means.
1. We are learning what a negative cosine ratio means with respect to the angle.
Cosine RuleCosine RuleNat 5
2. 2. Solve Solve REAL LIFE problems problems that involve finding an that involve finding an angle of a triangle.angle of a triangle.
C
B
A19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Cosine RuleCosine Rulew
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Nat 5
a
b
c
The Cosine Rule can be used with ANY triangle as long as we have been given enough information.
Works for any Triangle
cos2 2 2a =b +c  2bc A
19 Apr 202319 Apr 2023 Created by Mr Lafferty Maths DeptCreated by Mr Lafferty Maths Dept
Cosine RuleCosine Rulew
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Nat 5
How to determine when to use the Cosine Rule.
Works for any Triangle
1. Do you know ALL the lengths.
2. Do you know 2 sides and the angle in between.
SASOR
If YES to any of the questions then Cosine Rule
Otherwise use the Sine Rule
Two questions
Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule
Consider the Cosine Rule again:We are going to change the subject of the formula to cos Ao
Turn the formula around:b2 + c2 – 2bc cos Ao = a2
Take b2 and c2 across.2bc cos Ao = a2 – b2 – c2
Divide by – 2 bc.2 2 2
cos2
o a b cA
bc
Divide top and bottom by 12 2 2
cos2
o b c aA
bc
You now have a formula for finding an angle if you know all three sides of the triangle.w
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Nat 5 Works for any Triangle
Write down the formula for cos Fo
2 2 2
cos2
od e f
Fde
Label and identify Fo and d , e and f.
Fo = ? f = 8e = 10d = 12
Substitute values into the formula.
2 2 212 10 8cos
2 12 10
oF
Calculate cos Fo .Cos Fo =0.75
Use cos1 0.75 to find FoFo = 41.4o
Example : Calculate the
unknown angle Fo .
Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule
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Nat 5 Works for any Triangle
E
D
F
e
d
f
Example : Find the unknown
Angle in the triangle:
Write down the formula.
2 2 2
cos2
o b c aA
bc
Identify the sides and angle.
Ao = yo a = 26 b = 15 c = 13
2 2 215 13 26cos
2 15 13oA
Find the value of cosAo
cosAo =  0.723 The negative tells you
the angle is obtuse.Ao = 136.3ow
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Nat 5
Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule
Works for any Triangle A
BC
c
a
b
created by Mr. Laffertycreated by Mr. Lafferty
Now try N5 TJ Ex20.2
Ch 20 (Page 200)
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Solving Trig EquationsSolving Trig Equations
National 5
created by Mr. Laffertycreated by Mr. Lafferty
StarterStarter
2
1. Make a the subject of the f ormula
5 = 10b + a
2. Use the quadratic f ormula to solve
x + 5x + 1
3. Sketch the f unction y = 4sin3x
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National 5
created by Mr. Laffertycreated by Mr. Lafferty
Learning IntentionLearning Intention Success CriteriaSuccess Criteria
1.1. Know and learn the two Know and learn the two special trig relationships.special trig relationships.
2. Apply them to solve problems.
1. To explain some special trig relationships
sin 2 xo + cos 2 xo = ?
and
tan xo and sin x cos x
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Solving Trig Equations Solving Trig Equations National 5
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Solving Trig EquationsSolving Trig Equations
Lets investigate
sin 2xo + cos 2 xo = ?
Calculate value for x = 10, 20, 50, 250
sin 2xo + cos 2 xo = 1
Learn !
National 5
sin 2xo = 1  cos 2 xo
cos2xo = 1  sin2 xo
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Solving Trig EquationsSolving Trig Equations
Lets investigate
tan xo
Calculate value for x = 10, 20, 50, 250
Learn !
sin xo cos xo
and
tan xo sin xo cos xo =
National 5
cos2xo = 2516
cos2xo = 1  sin2xo
Given that sin xo = .35
Find cos xo .
cos2xo = 1  35(
(2
cos2xo = 1  925
√ cosxo = 54
sin2xo = 10064
sin2xo = 1  cos2xo
Given that cos xo = .6
10Find sin xo and tanxo.
(
(
sin2xo = 1  610
2
sin2xo = 1  10036
√ sinxo = 108
tan xo sin xo cos xo =
810
610
=
8
6=
4
3=tan xo
= sinxsin2x + sinxcos2x
= sinx(sin2x + sinxcos2x)
(sin2x + cos2x) = 1
LHS
= sinx = RHS
sinx cosx
cosx
sinx =
sinx
sinx =
= 1
LHS 1 – sin2A = cos2A
sin2A
cos2A =
= tan2A = RHS
created by Mr. Laffertycreated by Mr. Lafferty
Now try N5 TJ Ex20.3
Ch 20 (Page 201)
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Solving Trig EquationsSolving Trig Equations
National 5
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Higher
Are you on Target !
• Update you log book
• Make sure you complete and correct
ALL of the Trigonometry questions in the past paper booklet.