Find the Exact Value sin((13pi)/3)

Problem

$\sin \frac{13\pi }{3}$

Solution

1. Identify the angle and determine its position by finding a coterminal angle within the interval $[0,2\pi )$

2. Subtract multiples of $2\pi$ (which is $\frac{6\pi }{3}$ from the given angle.

$\frac{13\pi }{3}-2\pi =\frac{13\pi }{3}-\frac{6\pi }{3}=\frac{7\pi }{3}$

3. Subtract another $2\pi$ to bring the angle within the standard range.

$\frac{7\pi }{3}-\frac{6\pi }{3}=\frac{\pi }{3}$

4. Evaluate the sine of the coterminal angle $\frac{\pi }{3}$ using the unit circle or special right triangles.

$\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}$

Final Answer

$\sin \frac{13\pi }{3}=\frac{\sqrt{3}}{2}$

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