# Solutions to Questions on Identities and the Unit Circle

The solutions to the questions on trigonometric identities and the unit circle are presented.

## Use the following general identities

1) cos (A + B) = cos A cos B - sin A sin B
2) cos (A - B) = cos A cos B + sin A sin B
3) sin(A + B) = sin A cos B + cos A sin B
4) sin(A - B) = sin A cos B - cos A sin B

## to verify the identities listed below.

1. cos(π - θ) = - cos θ
2. sin(π - θ) = sin θ
3. cos(π + θ) = - cos θ
4. sin(π + θ) = - sin θ
5. cos(2π - θ) = cos θ
6. sin(2π - θ) = - sin θ
7. cos(- θ) = cos θ
8. sin( - θ) = - sin θ
9. cos(π/2 - θ) = sin θ
10. sin(π/2 - θ) = cos θ

## Solutions

1. Use the general identity cos (A - B) = cos A cos B + sin A sin B to expand cos(π - θ) as follows:
cos(π - θ) = cos π cos θ + sin π sin θ
Use cos π = - 1 and sin π = 0 to simplify the above to
= - cos θ)

2. Use the general identity sin(A - B) = sin A cos B - cos A sin B to expand sin(π - θ) as follows:
sin(π - θ) = sin π cos θ - cos π sin θ
Use sin π = 0 and cos π = - 1 to simplify the above to
= sin θ

3. Use the general identity cos (A + B) = cos A cos B - sin A sin B to expand cos(π + θ) as follows:
cos(π + θ) = cos π cos θ - sin π sin θ
Use cos π = - 1 and sin π = 0 to simplify the above to
= - cos θ)

4. Use the general identity sin(A + B) = sin A cos B + cos A sin B to expand sin(π + θ) as follows:
sin(π + θ) = sin π cos θ + cos π sin θ
Use sin π = 0 and cos π = - 1 to simplify the above to
= - sin θ

5. Use the general identity cos (A - B) = cos A cos B + sin A sin B to expand cos(2π - θ) as follows:
cos(2π - θ) = cos 2π cos θ + sin 2π sin θ
Use cos 2π = 1 and sin 2π = 0 to simplify the above to
= cos θ)

6. Use the general identity sin(A - B) = sin A cos B - cos A sin B to expand sin(2π - θ) as follows:
sin(2π - θ) = sin 2π cos θ - cos 2π sin θ
Use sin 2π = 0 and cos 2π = 1 to simplify the above to
= - sin θ

7. We first write the left side of the identity to verify cos(- θ) = cos θ as follows:
cos(- θ) = cos(0 - θ)
We then use the general identity cos (A - B) = cos A cos B + sin A sin B to expand cos(0 - θ) as follows:
cos(- θ) = cos(0 - θ) = cos 0 cos θ + sin 0 sin θ
Use cos 0 = 1 and sin 0 = 0 to simplify the above to
= cos θ)

8. We first write the left side of the given identity to verify sin( - θ) = - sin θ as follows:
sin( - θ) = sin (0 - θ)
We then use the general identity sin(A - B) = sin A cos B - cos A sin B to expand sin(0 - θ) as follows:
sin( - θ) = sin(0 - θ) = sin 0 cos θ - cos 0 sin θ
Use sin 0 = 0 and cos 0 = 1 to simplify the above to
= - sin θ

9. Use the general identity cos (A - B) = cos A cos B + sin A sin B to expand cos(π/2 - θ) as follows:
cos(π/2 - θ) = cos π/2 cos θ + sin π/2 sin θ
Use cos π/2 = 0 and sin π/2 = 1 to simplify the above to
= sin θ)

10. Use the general identity sin(A - B) = sin A cos B - cos A sin B to expand sin(π/2 - θ) as follows:
sin(π/2- θ) = sin π/2 cos θ - cos π/2 sin θ
Use sin π/2 = 1 and cos π/2 = 0 to simplify the above to
= cos θ