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Chapter 5: Problem 7

Evaluate \(\sin ^{-1}\left(\sin \frac{2 \pi}{7}\right)\)

Short Answer

Expert verified
The short answer is: \(\sin ^{-1}\left(\sin \frac{2 \pi}{7}\right) = \frac{2\pi}{7}\).

Step by step solution

01

Check the range of the angle

Check whether the given angle \(\frac{2 \pi}{7}\) lies in the range of the inverse sine function \(-\frac{1}{2} \pi \le x \le \frac{1}{2} \pi\). In this case, we have \(0 \le \frac{2 \pi}{7} \le \frac{1}{2} \pi\), since it is a positive angle smaller than \(\pi\) but larger than 0.
02

Find the sine of the angle

Calculate the sine of the given angle, which is \(\sin\left(\frac{2 \pi}{7}\right)\). This value won't simplify however it represents the output of the sine function for the given angle.
03

Evaluate the inverse sine of the sine value

To find the inverse sine of the value we found in step 2, simply calculate \(\sin ^{-1}\left(\sin\frac{2 \pi}{7}\right)\). Since the angle \(\frac{2 \pi}{7}\) is within the range of the inverse sine function, we don't need to adjust the angle any further. Therefore, the evaluation of the expression is \(\sin ^{-1}\left(\sin \frac{2 \pi}{7}\right) = \frac{2\pi}{7}\).

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Most popular questions from this chapter

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 4 and 5 in Section 6.3.] $$ \tan \frac{3 \pi}{8} $$

Suppose \(t\) is such that \(\tan ^{-1} t=\frac{3 \pi}{7}\). Evaluate the following: (a) \(\tan ^{-1} \frac{1}{t}\) (c) \(\tan ^{-1}\left(-\frac{1}{t}\right)\) (b) \(\tan ^{-1}(-t)\)

Suppose \(u\) and \(v\) are in the interval \(\left(\frac{\pi}{2}, \pi\right),\) with \(\tan u=-2\) and \(\tan v=-3\) Find exact expressions for the indicated quantities. $$ \cos v $$

Is sine an even function, an odd function, or neither?

Find exact expressions for the indicated quantities, given that $$ \cos \frac{\pi}{12}=\frac{\sqrt{2+\sqrt{3}}}{2} \text { and } \sin \frac{\pi}{8}=\frac{\sqrt{2-\sqrt{2}}}{2} $$ [These values for \(\cos \frac{\pi}{12}\) and \(\sin \frac{\pi}{8}\) will be derived in Examples 4 and 5 in Section 6.3.] $$ \sin \left(-\frac{5 \pi}{12}\right) $$
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