7. Evaluate each expression using the unit circle. Find the exact value. Show all work.
Write the ratio in reduced fractional form with a rationalized denominator if necessary.
a. sin 150° cos(−300°) − sin 810° cos 240°
To evaluate each of these expressions using the unit circle, we need to find the exact values of sine and cosine for the corresponding angles.
a. sin 150° cos(−300°) − sin 810° cos 240°
sin 150° = sin(180° - 30°) = sin 30° = 0.5
cos(-300°) = cos(360° - 300°) = cos 60° = 0.5
sin 810° = sin(720° + 90°) = sin 90° = 1
cos 240° = cos(180° + 60°) = -cos 60° = -0.5
Therefore, the expression becomes:
0.5 * 0.5 - 1 * (-0.5)
= 0.25 + 0.5
= 0.75
So, sin 150° cos(−300°) − sin 810° cos 240° = 0.75.