Question

find the exact value.

$$\sin \frac { 7 \pi } { 4 }$$

type + or -

$$\frac { \sqrt { \square } } { \square }$$

Explanation:

Step1: Determine the quadrant

$\frac{7\pi}{4}$ is in the fourth quadrant. In the fourth quadrant, the sine function is negative.

Step2: Find the reference angle

The reference angle for $\frac{7\pi}{4}$ is $2\pi-\frac{7\pi}{4}=\frac{\pi}{4}$.

Step3: Evaluate the sine of the reference angle

We know that $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$.

Step4: Apply the sign based on the quadrant

Since $\frac{7\pi}{4}$ is in the fourth quadrant (where $\sin\theta<0$), $\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}$.

Answer:

$-\frac{\sqrt{2}}{2}$