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 expression, we can use the unit circle and the trigonometric identities:
1. sin 150° = sin (180° - 30°) = sin 180° cos 30° - cos 180° sin 30°
Since sin 180° = 0 and cos 180° = -1, this simplifies to -sin 30° = -0.5
2. cos(-300°) = cos(360° - 300°) = cos 60° = 0.5
3. sin 810° = sin (720° + 90°) = sin 90° = 1
cos 240° = cos (180° + 60°) = -cos 60° = -0.5
Putting it all together:
sin 150° cos(-300°) - sin 810° cos 240°
= (-0.5 * 0.5) - (1 * -0.5)
= -0.25 + 0.5
= 0.25
Therefore, the exact value of the expression is 0.25.