Use the reference angle to find the exact value of the following expression. Show all your work step by step.
sin (-150°)
To find the exact value of sin(-150°), we can use the fact that the sine function has a period of 360° and that sin(x) = sin(180° - x) for any angle x.
Step 1: Find the reference angle
The reference angle is the acute angle between the terminal side of the angle and the x-axis. To find it, we take the absolute value of the angle and find the difference between 180° and the result:
|(-150°)| = 150°
Reference angle = 180° - 150° = 30°
Step 2: Determine the quadrant
Since the angle -150° is in the third quadrant (between 180° and 270°), the sine value will be negative.
Step 3: Evaluate sin(30°)
The exact value of sin(30°) is 1/2.
Step 4: Apply the symmetry property
Since -150° is in the third quadrant, the sine value will be opposite in sign to the corresponding reference angle of 30°. Therefore, sin(-150°) = -sin(30°).
Step 5: Substitute the value of sin(30°)
sin(-150°) = -(1/2) = -1/2
Therefore, the exact value of sin(-150°) is -1/2.