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Chapter 4: Problem 34

Evaluate the trigonometric function using its period as an aid. $$\sin \frac{9 \pi}{4}$$

Short Answer

Expert verified
The value of \(\sin\frac{9 \pi}{4}\) is \(\sqrt{2}/2\).

Step by step solution

01

Identify the period of the trigonometric function

First, identify a period of the trigonometric function. For the sine function, the period is \(2\pi\).
02

Find the equivalent angle within the period

Since \(9 \pi/4\) is more than \(2\pi\), we start 'over' with the periods. Any time an angle is a multiple of \(2\pi\), we are back to where we started. Find the equivalent angle within the first period by subtracting or adding multiples of \(2\pi\) from the given angle, until you reach the angle within the range [-\(\pi\), \(\pi\)]. Here, subtract \(2\pi\) from \(9 \pi/4\), which gives \(9 \pi/4 - 2\pi = 9 \pi/4 - 8 \pi/4= \pi/4\).
03

Compute the sine of the equivalent angle

Compute the sine of the angle \(\frac{\pi}{4}\) derived from the previous step. From the unit circle or right triangle definitions, we know that \(\sin(\pi / 4) = \sqrt{2}/2\).

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