**Trigonometric Function Evaluation****Question:** Give the exact value of \( \sin{\left(\frac{15\pi}{4}\right)} \).**Answer choices:**- \( \frac{1}{2} \)- undefined- \( \frac{-\sqrt{2}}{2} \)- \(-1\)- \(0\)- \( \frac{-\sqrt{3}}{2} \)- \( \frac{-1}{2} \)- \(1\)- \( \frac{\sqrt{3}}{2} \)- \( \frac{\sqrt{2}}{2} \)**Explanation:**To find the exact value of \( \sin{\left(\frac{15\pi}{4}\right)} \), we can first convert the angle to a simpler equivalent angle within the range \(0\) to \(2\pi \).### Step 1: Simplify the Angle\[ \frac{15\pi}{4} \] can be simplified by subtracting \(2\pi\) multiple times (since \(2\pi\) is a complete circle).\[ \frac{15\pi}{4} - 2\pi = \frac{15\pi}{4} - \frac{8\pi}{4} = \frac{7\pi}{4} \]The angle \( \frac{7\pi}{4} \) falls within the interval \(0\) to \(2\pi \), which makes it easier to evaluate.### Step 2: Identify the QuadrantThe angle \( \frac{7\pi}{4} \) is in the fourth quadrant, where sine is negative.### Step 3: Find the Reference AngleThe reference angle for \( \frac{7\pi}{4} \) is \( 2\pi - \frac{7\pi}{4} = \frac{8\pi}{4} - \frac{7\pi}{4} = \frac{\pi}{4} \).### Step 4: Evaluate the Sine FunctionUsing the reference angle \(\frac{\pi}{4}\) and knowing the sine values for standard angles:\[ \sin{\left(\frac{\pi}{4}\right)} = \frac{\sqrt{2}}{2} \]Since \( \frac{7\pi}{4

Name: **Trigonometric Function Evaluation****Question:** Give the exact val
Uploaded: 2024-03-20T06:46:55.000Z
Channel: Bartleby
Description: **Trigonometric Function Evaluation****Question:** Give the exact value of \( \sin{\left(\frac{15\pi}{4}\right)} \).**Answer choices:**- \( \frac{1}{2} \)- undefined- \( \frac{-\sqrt{2}}{2} \)- \(-1\)- \(0\)- \( \frac{-\sqrt{3}}{2} \)- \( \frac{-1}{