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Chapter 5: Problem 23

Suppose \(\frac{\pi}{2}<\theta<\pi\) and \(\sin \theta=\frac{2}{9} .\) Evaluate \(\cos \theta\).

Short Answer

Expert verified
The value of \(\cos\theta\) is \(-\sqrt{\frac{77}{81}}\).

Step by step solution

01

Recognize the quadrant and quadrant properties

Since \(\frac{\pi}{2}<\theta<\pi\), the angle 胃 is in the second quadrant. In the second quadrant, sine is positive and cosine is negative. This information will help us to determine the correct sign for our answer.
02

Apply the Pythagorean identity

Use the identity \(\sin^2\theta + \cos^2\theta = 1\) to find the value of \(\cos^2\theta\). We are given \(\sin\theta = \frac{2}{9}\), so we can substitute this value into the identity: \(\left(\frac{2}{9}\right)^2 + \cos^2\theta = 1\)
03

Solve for \(\cos^2\theta\)

Calculate \(\left(\frac{2}{9}\right)^2\) and rearrange the equation to find the value of \(\cos^2\theta\): \(\frac{4}{81} + \cos^2\theta = 1\) \(\cos^2\theta = 1 - \frac{4}{81}\) \(\cos^2\theta = \frac{77}{81}\)
04

Find the value of \(\cos\theta\)

Since we know that cosine is negative in the second quadrant, we can determine the value of \(\cos\theta\) by taking the negative square root of \(\cos^2\theta\): \(\cos\theta = -\sqrt{\frac{77}{81}}\) So, the value of \(\cos\theta\) is \(-\sqrt{\frac{77}{81}}\).

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