Problem 41 - Precalculus Essentials

Question

Find the exact value of each trigonometric function. Do not use a calculator. $$\sin \left(-\frac{9 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The value of \(\sin\left(-\frac{9 \pi}{4}\right)\) equals \(\sqrt{2}/2\)

Step 1: Convert the angle to positive

The given angle is \(-\frac{9 \pi}{4}\) which is negative. It can be converted into positive by adding \(2\pi\) (which is the same as \(8\pi/4\)) to the given angle, thus obtaining \(-\frac{9 \pi}{4} + \frac{8 \pi}{4} = -\frac{\pi}{4}\)

Step 2: Apply Co-terminal angle concept

In order to find the sin of a negative angle, we use the co-terminal angle concept in trigonometry. The co-terminal of \(-\pi/4\) is \(2\pi - \pi/4 = 7\pi/4\)

Step 3: Find the value of the sine function

We then take the sine of the equivalent positive angle. The sine of \(7\pi/4\) equals \(\sqrt{2}/2\). We use basic knowledge of the sine function and that the sine of \(7\pi/4\) is equal to the sine of \(\pi/4\), which is \(\sqrt{2}/2\)