Sum and Difference Formulas
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Transcript of Sum and Difference Formulas
Sum and Difference Formulas
Section 5.4
Exploration: Are the following functions equal?
a) Y = Cos (x + 2) b) Y = Cos x + Cos 2
How can we determine if they are equal by looking at their
graphs?
Graph them using your calculator.
Explorationa) Y = Cos (x + 2) b) Y = Cos x
+ Cos 2
Y = Cos (x + 2)Y = Cos x + Cos 2
Y = Cos x + Cos 2Y = Cos (x + 2)
Sum and Difference Formulas
Sin (u + v) =
Sin (u – v) =
Cos (u + v) =
Cos (u – v) =
Sin u Cos v + Cos u Sin v
Sin u Cos v – Cos u Sin v
Cos u Cos v – Sin u Sin v
Cos u Cos v + Sin u Sin v
Sum and Difference Formulas
Tan (u + v) =
Tan (u – v) =
Tan u + Tan v1 - Tan u Tan v
Tan u - Tan v1 + Tan u Tan v
Sum and Difference Formulas Before we continue, think about all of the
angles you can find a trig function without using a calculator:
Choose any 2 of these and a trig function:
Sum and Difference Formulas
Sum and Difference Formulas To find the trig function of an angle using the
formulas:
1) Find 2 angles whose sum or difference is equal to the angle you are trying to evaluate
2) Put the two angles into the appropriate formula
3) Evaluate the trig functions of the angles you know
4) Simplify
Sum and Difference Formulas Evaluate: Sin 15º
What two angles have a sum or difference of 15º?→ 45º - 30º
Put these two angles in the appropriate formula:→ Sin (45º - 30º)= Sin 45º Cos 30º - Cos 45º Sin 30º
Sum and Difference Formulas Sin 45º Cos 30º - Cos 45º Sin 30º
Evaluate the trig functions
22
23
22
21
Simplify
46
42
426
Sum and Difference Formulas
127 Cos
4
3
127
)4
3
( Cos 127 Cos
4Sin
3Sin -
4 Cos
3 Cos
127 Cos
Sum and Difference Formulas
4Sin
3Sin -
4 Cos
3 Cos
127 Cos
21
22
23
22
462
Sum and Difference Formulas Evaluate the following functions.
a)
b)
o75 Cos
12Sin
426
426
Sum and Difference Formulas
o75 Cos = Cos 45º Cos 30º - Sin 45º Sin 30º
22
23
22
21
426
Sum and Difference Formulas
12
Sin
23
22
21
22
426
4Sin
3 Cos -
4 Cos
3Sin
Sum and Difference Formulas
Section 5.4
Sum and Difference Formulas Evaluate the following functions.
a)
b)
o105- Cos
125Tan
462
23
Sum and Difference Formulas
o105- Cos = Cos 150º Cos 45º + Sin 150º Sin 45º
23-
22
21
22
462
Sum and Difference Formulas
125Tan )
46(Tan
)4
Tan()6
(Tan -1
)4
Tan()6
(Tan
33 - 1
133
333
333
Sum and Difference Formulas
125Tan
3-3 33
33 33
6 3612
32
Sum and Difference Formulas Yesterday:
Used the formulas to evaluate trig functions of different angles
Worked with both radians and degrees
Today Use the formulas to simplify longer expressions Use the formulas to evaluate expressions from
triangles Use the formulas to create algebraic expressions
Sum and Difference Formulas Find the exact value of the following
expression:
Cos 78ºCos18º + Sin 78ºSin18º
What formula is being used here?→ Cos (u – v)
Re-write the expression using the formula→ Cos (78º – 18º)
= Cos 60º = ½
= ½
Sum and Difference Formulas Use the sum and difference formulas to
evaluate the following:oooo 12Sin 42 Cos - 12 Cos42Sin a)
5Sin
7Sin -
5 Cos
7 Cos b)
o30Sin 21
)57
( Cos
3512 Cos
Sum and Difference Formulas Find the exact value of the Cos (u – v) using the
given information:
Sin u = Cos v = Both u and v and in quadrant III 25
7
54
When you are given 2 different criteria, you must draw 2 different triangles
u-7
25
-24- 3
5
- 4v
Sum and Difference Formulas
u-7
25
-24- 3
5
- 4v
Cos (u – v) = Cos u Cos v + Sin u Sin v
2524
54
25
7
53
12596
12521
125117
Sum and Difference Formulas Find the exact value of the trig functions given
the following information:
Tan u = Csc v =
and both u and v are in quadrant IV.
Find a) Sin (u + v) b) Sec (u – v)
c) Cot (u – v)
43
5
13
5663
6365
6556
Sum and Difference Formulas
u-3
5
4v
-513
12
Sin (u + v) = Sin u Cos v + Cos u Sin v
53
1312
54
13
5
6536
6520
6556
Sum and Difference Formulas
u-3
5
4v
-513
12
Sec (u - v) = Cos u Cos v + Sin u Sin v
54
1312
53
13
5
6548
6515
6563
1 ÷ Cos (u - v)Cos (u - v) =
6365
Sum and Difference Formulas
u-3
5
4v
-513
12
Cot (u - v) =
125
431
125
43
1 ÷ Tan (u - v)
Tan (u - v) = Tan vu Tan 1Tan v -u Tan
4815
4848
4820
4836
48634856
6356
5663
Sum and Difference Formulas Lastly, we would like to apply the process
used in drawing triangles to create algebraic expressions.
Same steps as before, just using variables instead of numbers.
Sum and Difference Formulas Write Cos (arcTan 1 + arcCos x) as an
algebraic statement.
→ What formula is being used?Cos (u + v)
u = arcTan 1 v = arcCos xTan u = 1 Cos v = x
→ Use this information to draw your triangles.
Sum and Difference FormulasTan u = 1 Cos v = x
u1
1
2
vx
1 21 x
Cos (u + v) = Cos u Cos v – Sin u Sin v
21
x2
1 21 x
2x
2
1 2x
21 2xx
Sum and Difference Formulas Write the trig expression as an algebraic
expression:
Sin (arcTan 2x – arcCos x)
Sin (u – v)u = arcTan 2x v = arcCos xTan u = 2x Cos v = x
Sum and Difference Formulas
u 2x
1
241 x
vx
1 21 x
Sin (u – v) = Sin u Cos v – Cos u Sin v
241
2
x
x
x
241
1
x 21 x
2
2
41
2
x
x
2
2
41
1
x
x
2
22
41
12
x
xx
Tan u = 2x Cos v = x
Sum and Difference Formulas
Section 5.4
Sum and Difference Formulas Write the trig expression as an algebraic
expression:
Cos (arcSin 3x + arcTan 2x)
Cos (u + v)u = arcSin 3x v = arcTan 2xSin u = 3x Tan v = 2x
Sum and Difference Formulas
u 3x1
291 xv
1
x2
Cos (u + v) = Cos u Cos v – Sin u Sin v291 x 241
1
x x3
241
2
x
x
2
2
41
91
x
x
2
2
416x
x
2
22
41
691
x
xx
Sin u = 3x Tan v = 2x241 x
Sum and Difference Formulas So far, in this section we have:
a) Used sum and difference formulas to evaluate trig functions of different angles
b) Recognized sum and difference formulas to simplify expressions
c) Used criteria to draw triangles and apply formulas
d) Create algebraic expressions
Lastly, we are going to simplify, verify, and solve equations
Sum and Difference Formulas Simplifying:
o Apply the formula
o Evaluate trig functions that you know
o Reduce the expression
Sum and Difference Formulas Simplify the following expressioni:
Sin (90º – x)
→ Sin 90º Cos x – Cos 90º Sin x
→ (1)(Cos x)- (0)(Sin x)= Cos x
Sum and Difference Formulas Simplify the following expressioni:
Cos (x + 3π)
→ Cos x Cos 3π – Sin x Sin 3π
→ (Cos x)(0)- (Sin x)(1)= Sin x
Sum and Difference Formulas Verifying
Same process and simplifying
You are given what the expression should simplify to
As before, only work with 1 side of the equal sign
Sum and Difference Formulas Verify the following identities:
a) Tan (π + x) = Tan x
b) Sin (x + y) Sin (x – y) = Cos² y – Cos² x
Sum and Difference FormulasTan (π + x) = Tan x
Tan x Tan 1Tan x Tan
Tan x (0) 1
Tan x 0
1Tan x
Tan x
Sum and Difference FormulasSin (x + y) Sin (x – y) = Cos² y – Cos² x
= (Sin x Cos y + Cos x Sin y)( Sin x Cos y – Sin x Cos y)
= Sin² x Cos² y - Cos² x Sin² y= (1 - Cos² x) Cos² y- Cos² x (1 – Cos² y)= Cos² y - Cos² x Cos² y- Cos² x + Cos² y Cos² x= Cos² y – Cos² x
Sum and Difference Formulas The last step in this section is using the sum
and difference formulas to solve equations.
Again, apply the formula, simplify, and now solve.
Sum and Difference Formulas
1 )4
-(x Cos - )4
(x Cos
4Sin Sin x
4 Cos x Cos
)4
Sin Sin x 4
Cos x (Cos -
4Sin Sin x
4 Cos x Cos
)4
Sin Sin x 4
Cos x Cos-
4Sin 2Sin x
1
Sum and Difference Formulas
1 )4
-(x Cos - )4
(x Cos
4Sin 2Sin x
1 1 Sin x 222
1 Sin x 2 2
1 - Sin x
x ,4
5 4
7
Sum and Difference Formulas
1 )3
(x Sin )3
(x Sin
3Sin x Cos
3 CosSin x
3
Sin x Cos 3
CosSin x
3 Cos2Sin x
1
Sum and Difference Formulas
1 Sin x 212
1 Sin x x
2
1 )3
(x Sin )3
(x Sin
3 Cos2Sin x
1