Addition or Subtraction Formula

Question

Use an addition or subtraction formula to complete the following two steps: write the expression as a trigonometric function of one number, and then find its exact value. Do not use a calculator

sin 122° cos 32° - cos 122° sin 32°

Answer 1

sin(122°) = sin(180° - 58°)

cos(32°) = cos(90° - 58°)

cos(122°) = cos(180° - 58°)

sin(32°) = sin(90° - 58°)

sin(122°)cos(32°) - cos(122°)sin(32°)

sin(180° - 58°)cos(90° - 58°) - cos(180° - 58°)sin(90° - 58°)

Using the sin and cos addition and subtraction identities:

sin(x – y) = sin(x)cos(y) – cos(x)sin(y)

cos(x – y) = cos(x)cos(y) + sin(x)sin(y)

The expression becomes:

[sin(180°)cos(58°) – cos(180°)sin(58°)][cos(90°)cos(58°) + sin(90°)sin(58°)] - [cos(180°)cos(58°) + sin(180°)sin(58°)][sin(90°)cos(58°) – cos(90°)sin(58°)] =

[(0)cos(58°) – (-1)sin(58°)][(0)cos(58°) + (1)sin(58°)] - [(-1)cos(58°) +(0)sin(58°)][(1)cos(58°) – (0)sin(58°)] =

sin(58°)sin(58°) - (-cos(58°)cos(58°) =

sin²(58°) + cos²(58°) = 1