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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

Solution

  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

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Updated: 9 March 2017 (A Dendane)

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