Question

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin (22𝜋/3)

Accepted Answer

Recognize that the angle given is in radians: $$\frac{22\pi}{3}. $$Since the sine function is periodic with period $$2\pi$$, we can reduce the angle by subtracting multiples of $$2\pi to $$find a coterminal angle within the standard interval $$[0, 2\pi). $$Calculate how many full $$2\pi$$ rotations fit into $$\frac{22\pi}{3}. $$Since $$2\pi = \frac{6\pi}{3}$$, divide $$\frac{22\pi}{3} by$$ $$\frac{6\pi}{3} to $$find the quotient: $$\frac{22\pi/3}{6\pi/3} = \frac{22}{6} = 3 + \frac{4}{6} = 3 + \frac{2}{3}. $$Subtract $$3 	imes 2\pi = 6\pi$$ from $$\frac{22\pi}{3} to $$find the coterminal angle: $$\frac{22\pi}{3} - 6\pi = \frac{22\pi}{3} - \frac{18\pi}{3} = \frac{4\pi}{3}. $$Now, evaluate $$\sin \left( \frac{4\pi}{3} ight). $$Recall that $$\frac{4\pi}{3} is in $$the third quadrant where sine is negative, and it corresponds to an angle of $$\pi + \frac{\pi}{3}. $$Use the sine addition formula or reference the unit circle to express $$\sin \left( \frac{4\pi}{3} ight) as$$ $$-\sin \left( \frac{\pi}{3} ight)$$, and recall that $$\sin \left( \frac{\pi}{3} ight) = \frac{\sqrt{3}}{2}.$$

In Exercises 49–59, find the exact value of each expression. Do not use a calculator. sin (22𝜋/3)

Verified video answer for a similar problem:

Key Concepts

Radian Measure and Angle Conversion

Periodic Properties of the Sine Function

Exact Values of Sine for Special Angles