1. Find the following: sin 30˚ (no calculator allowed)
sin 75˚ (use a calculator)
Is sin 30˚ + sin 45˚ = sin 75˚?
What two unit circle values combine to be 165˚?
What two unit circle values combine to be 105˚?

2. Find the following: sin 30˚ (no calculator allowed) = ½
sin 75˚ (use a calculator)
Is sin 30˚ + sin 45˚ = sin 75˚?
What two unit circle angles combine to be 165˚?
What two unit circle angles combine to be 105˚?
= ½ = =
No, the two are not equal
½ ≠
120 + 45
135 – 30 or 60 + 45

3. Difference Identities
Chapter 5 Trigonometric Equations
5.4
Using Sum and
Difference Identities
MATHPOWERTM 12, WESTERN EDITION
5.4.1

4. Sum and Difference Identities
sin(A + B) = sin A cos B + cos A sin B
sin(A - B) = sin A cos B - cos A sin B
cos(A + B) = cos A cos B - sin A sin B
cos(A - B) = cos A cos B + sin A sin B
5.5.2

5. 1. Find the exact value for sin 750.
Finding Exact Values
1. Find the exact value for sin 750.
Think of the angle measures that produce exact values:
300, 450, and 600.
Use the sum and difference identities.
Which angles, used in combination of addition
or subtraction, would give a result of 750?
sin 750 = sin( )
= sin 300 cos cos 300 sin 450
5.5.4

6. 2. Find the exact value for cos 150. cos 150 = cos(450 - 300)
Finding Exact Values
2. Find the exact value for cos 150.
cos 150 = cos( )
= cos 450 cos sin 450 sin300
3. Find the exact value for
5.5.5

7. 4. Find the exact value for tan 1650. tan 1650 = tan(1350 + 300)
Finding Exact Values
4. Find the exact value for tan 1650.
tan 1650 = tan( )

8. Rationalize the denominator
Multiply top and bottom by the conjugate of the bottom

9. 4. Find the exact value for tan 1650 (continued)
Finding Exact Values
4. Find the exact value for tan 1650 (continued)
tan 1650
Rationalize the denominator

10. Assignment

11. Simplifying Trigonometric Expressions1.
Express cos 1000 cos sin 800 sin 1000 as a
trig function of a single angle.
This function has the same pattern as cos (A - B),
with A = 1000 and B = 800.
cos 100 cos 80 + sin 80 sin 100 = cos( )
= cos 200 2.
Express
as a single trig function.
This function has the same pattern as sin(A - B), with
5.5.3

12. Using the Sum and Difference Identities
Prove
L.S. = R.S.
5.5.6

13. Using the Sum and Difference Identities
x = 3
r = 5 θ 5 3
y = ± 4
sinθ = 4/5
5.5.7

14. Using the Sum and Difference Identities
A B x y r 4 2 3 3 5
5.5.8

15. Assignment Complete #’s 3-12 on worksheet Answers: – sin x Hints:
cos x
sin x
– tan x + 2 sin x
Hints:
345 =
195 = 240 – 45

16. Identities
Prove
L.S. = R.S.
5.5.16

17. Double-Angle Identities
The identities for the sine and cosine of the sum of two
numbers can be used, when the two numbers A and B
are equal, to develop the identities for sin 2A and cos 2A.
sin 2A = sin (A + A)
= sin A cos A + cos A sin A
= 2 sin A cos A
cos 2A = cos (A + A)
= cos A cos A - sin A sin A
= cos2 A - sin2A
Identities for sin 2x and cos 2x:
sin 2x = 2sin x cos x
cos 2x = cos2x - sin2x
cos 2x = 2cos2x - 1
cos 2x = 1 - 2sin2x
5.5.9

18. Double-Angle Identities
Express each in terms of a single trig function.
a) 2 sin 0.45 cos 0.45
b) cos2 5 - sin2 5
sin 2x = 2sin x cos x
sin 2(0.45) = 2sin 0.45 cos 0.45
sin 0.9 = 2sin 0.45 cos 0.45
cos 2x = cos2 x - sin2 x
cos 2(5) = cos2 5 - sin2 5
cos 10 = cos2 5 - sin2 5
Find the value of cos 2x for x = 0.69.
cos 2x = cos2 x - sin2 x
cos 2(0.69) = cos sin2 0.69
cos 2x =
5.5.10

19. Double-Angle Identities
Verify the identity
L.S = R.S.
5.5.11

20. Double-Angle Identities
Verify the identity
L.S = R.S.
5.5.12

21. Double-Angle Equationsp 2p
y = cos 2A
5.5.13

22. Double-Angle Equations
y = sin 2A
5.5.14

23. Rewrite f(x) as a single trig function: f(x) = sin 2x
Using Technology
Graph the function
and predict the period.
The period is p.
Rewrite f(x) as a single trig function:
f(x) = sin 2x
5.5.15

24. Applying Skills to Solve a Problem
The horizontal distance that a soccer ball will travel, when
kicked at an angle q, is given by ,
where d is the horizontal distance in metres, v0 is the initial
velocity in metres per second, and g is the acceleration due
to gravity, which is 9.81 m/s2.
a) Rewrite the expression as a sine function.
Use the identity sin 2A = 2sin A cos A:
5.5.17

25. Applying Skills to Solve a Problem [cont’d]
b) Find the distance when the initial velocity is 20 m/s.
From the graph, the maximum
distance occurs when q =
The maximum distance is m.
The graph of sin q reaches its
maximum when
Sin 2q will reach a maximum when
Distance
Angle q
5.5.18

26. Assignment Suggested Questions: Pages 272-274 A 1-16, 25-35 odd
B , 37-40, 43,
47, 52
5.5.19