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Chapter 5: Problem 10
What angle corresponds to a circular arc on the unit circle with length \(\frac{\pi}{6} ?\)
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Show that $$ \sin (\pi-\theta)=\sin \theta $$ for every angle \(\theta\).
Show that $$ \begin{aligned} \cos ^{3} \theta+\cos ^{2} \theta \sin \theta &+\cos \theta \sin ^{2} \theta+\sin ^{3} \theta \\ &=\cos \theta+\sin \theta \end{aligned} $$ for every number \(\theta\). [Hint: Try replacing the \(\cos ^{2} \theta\) term above with \(1-\sin ^{2} \theta\) and replacing the \(\sin ^{2} \theta\) term above with \(1-\cos ^{2} \theta .\)
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. $$ \cos (-u) $$
Evaluate \(\sin \left(\tan ^{-1}(-9)\right)\)
Evaluate \(\tan ^{-1}\left(\tan \frac{11 \pi}{5}\right)\).
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