Use the steps to determine the exact value of \( \sin (-135)^{\circ} \). Identify the reference angle, \( \theta \). \( \theta=45 \) CompLETE Find the value of the function for \( \theta \). \( \sin \left(45^{\circ}\right)= \) ? \( 1 / 2 \) \( \sqrt{2} / 2 \) \( \sqrt{3} / 2 \) DONEI

To determine the exact value of \( \sin(-135^\circ) \), we will follow the steps outlined: 1. **Identify the reference angle**: - The angle \( -135^\circ \) is in the fourth quadrant when considering the unit circle. To find the reference angle, we can convert it to a positive angle by adding \( 360^\circ \): \[ -135^\circ + 360^\circ = 225^\circ \] - The reference angle \( \theta \) is calculated as: \[ \theta = 225^\circ - 180^\circ = 45^\circ \] 2. **Find the value of the function for \( \theta \)**: - We need to find \( \sin(45^\circ) \). The sine of \( 45^\circ \) is a well-known value: \[ \sin(45^\circ) = \frac{\sqrt{2}}{2} \] 3. **Determine the sign of the sine function**: - Since \( -135^\circ \) is in the third quadrant (where sine is negative), we have: \[ \sin(-135^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2} \] Thus, the exact value of \( \sin(-135^\circ) \) is: \[ \sin(-135^\circ) = -\frac{\sqrt{2}}{2} \] To summarize: - Reference angle \( \theta = 45^\circ \) - \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \) - Therefore, \( \sin(-135^\circ) = -\frac{\sqrt{2}}{2} \)