Use the reference angle to find the exact value of the following expression. Show all your work step by step.
sin (-135°)
To find the exact value of sin (-135°) using the reference angle, we first need to determine the reference angle for -135° within the unit circle.
-135° is located in the third quadrant of the unit circle, which is right above the x-axis. To find the reference angle, we will subtract -135° from 180°.
Reference angle = 180° - (-135°)
Reference angle = 180° + 135°
Reference angle = 315°
Now that we have the reference angle of 315°, we can determine the exact value of sin (-135°) using the reference angle:
sin (-135°) = sin (360° - 135°)
sin (-135°) = sin (225°)
We know that the sine function has a positive value in the second quadrant, so we need to determine the value of sin (225°) in the second quadrant.
sin (225°) = -sin (180° + 45°)
sin (225°) = -sin (225° - 180°)
sin (225°) = -sin (45°)
We know that the sine function has a positive value for angles less than 90°, so we can write:
sin (45°) = sin (45°)
Since the sine function is symmetric about the y-axis, we can say that:
sin (45°) = sin (-45°)
Therefore:
sin (-135°) = -sin (-45°)
Next, we need to determine the value of sin (-45°). Since -45° is located in the fourth quadrant, we can determine its reference angle by subtracting -45° from 360°.
Reference angle = 360° - (-45°)
Reference angle = 360° + 45°
Reference angle = 405°
Now, using the reference angle of 405°, we can determine the exact value of sin (-45°) using the reference angle:
sin (-45°) = sin (360° - 45°)
sin (-45°) = sin (315°)
We know that the sine function has a negative value in the fourth quadrant, so we need to determine the value of sin (315°) in the fourth quadrant.
sin (315°) = -sin (360° - 315°)
sin (315°) = -sin (45°)
We already determined that sin (45°) = sin (-45°), so we can substitute this value:
sin (315°) = -sin (-45°)
Therefore, we can conclude that:
sin (-135°) = -sin (-45°)
Finally, we substitute sin (-45°) with its exact value:
sin (-135°) = -(-1/√2)
sin (-135°) = 1/√2
Therefore, the exact value of sin (-135°) using the reference angle is 1/√2.