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Chapter 4: Problem 36

Evaluate the trigonometric function using its period as an aid. $$\sin \left(-\frac{8 \pi}{3}\right)$$

Short Answer

Expert verified
The answer is \(\frac{\sqrt{3}}{2}\).

Step by step solution

01

Identify the period of the function

The trigonometric function used is the sine function, which has a period of \(2\pi\). Sine function is periodic, which means if you add or subtract any multiple of \(2\pi\) to/from the argument of the function, it will not change the result.
02

Relate the given angle to a standard angle

The given angle is \(-\frac{8 \pi}{3}\). To bring this angle within the period \(2\pi\), add a multiple of \(2\pi\) to the given angle so it falls within a standard range. \(-\frac{8 \pi}{3} + 3\pi = -\frac{8 \pi}{3} + \frac{9 \pi}{3} = \frac{\pi}{3}\). Here, \(\frac{\pi}{3}\) lies in the standard range of the sine function.
03

Evaluate the sine value

Now, the value of the sine function at the adjusted angle \(\sin(\frac{\pi}{3})\) is \(\frac{\sqrt{3}}{2}\), which will also be the value of \(\sin(-\frac{8 \pi}{3})\) due to the periodic nature of the sine function.

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