Download Ch. 4 – Trigonometric Functions

Ch. 5 – Analytic Trigonometry
5.4 – Sum and Difference Formulas


Angle sum and difference formulas can be used to find
exact values of trig functions not on the unit circle.
Ex: Find the exact value of cos75°.
◦
◦
◦
◦
◦
75° isn’t on the unit circle, but 30° and 45° are, and 75 = 30 + 45!
Instead of finding cos75°, find cos(30° + 45°)!
Look up your angle sum formula…
cos(a + b) = cosa cosb – sina sinb
cos(30 + 45) = cos30 cos45 – sin30 sin45
3 2 1 2


2 2 2 2
6
2


4
4
6 2

4
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Ex: Find the exact value of sin(π/12).
◦ Convert to degrees if you like!
◦ Find angles that either add or subtract to π/12…
  
  
3 4
◦ Look up your angle sum formula…
◦ sin(a – b) = sina cosb – sinb cosa




  
sin     sin cos  sin cos
3
4
4
3
3 4
3 2
21


2 2
2 2
6
2


4
4
6 2

4
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Ex: Write cos(arctan 1 + arccos x) as an algebraic expression.
◦ This is like an angle sum problem!
◦ Draw a right triangle for each inverse trig function. Label the angles
differently.
◦ Use Pythagorean Thm. to complete the triangles…
2
1
1  x2
1
u
v
x
1
◦ We’re just finding cos(u + v) = cosu cosv – sinu sinv
1 x 1 1  x2


1
21
2
x  1  x2

2
x
1  x2


2
2
I’ll let you leave the radical in the denominator
only for these algebraic answers.
Find the exact value of cos(-15°).
1.
6 2
4
2.
3.
6 2
2
4.
5.
6 2
2
6 2
4
 6 2
4
0%
0%
0%
0%
0%
Simplify tan(θ – 3π).
1.
tan 
2.
3.
tan 
1  tan 
4.
5.
undefined
tan 
1  tan 
tan   1
1  tan 
0%
0%
0%
0%
0%
Simplify sin(x + x).
1.
1
2
2
2.
sin x cos x
3.
sin x  cos x
4.
0
5.
2
2
2sin x cos x
0%
0%
0%
0%
0%
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Ex: Angles u and v both lie in quadrant I. If sinu = 12/13 and
cosv = 3/5, find sin(u+v).
◦ No angles are given this time, just trig ratios…and not even trig ratios
from the unit circle!
◦ Draw some triangles:
5
13
12
u
4
v
5
3
◦ Use Pythagorean Theorem to get the other side lengths
◦ To get sin(u+v), use an angle sum formula:
sin(u  v)  sin u cos v  sin v cos u
36
20
12 3
4 5
56





65
65
13 5
5 13
65

Ex: Angles u and v both lie in quadrant II. If tanu = -7/24 and
sinv = 8/17, find cos(u-v).
◦ Draw some triangles, but think…the adjacent side should be negative in
both triangles because both angles lie in the 2nd quadrant:
25
u
17
7
v
24
15
◦ Use Pythagorean Theorem to get the other side lengths
◦ To get cos(u-v), use an angle sum formula:
cos(u  v)  cos u cos v  sin u sin v
24 15
7 8


25 17
25 17
360
56
416



425
425 425
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