91Ó°ÊÓ

In Exercises \(87-106\), find the exact value or state that it is undefined. $$ \operatorname{arccot}\left(\cot \left(\frac{2 \pi}{3}\right)\right) $$

Verify the identity. Assume that all quantities are defined. $$ \frac{1-\cot (\theta)}{1+\cot (\theta)}=\frac{\tan (\theta)-1}{\tan (\theta)+1} $$

Verify the identity. Assume that all quantities are defined. $$ \frac{1-\tan (\theta)}{1+\tan (\theta)}=\frac{\cos (\theta)-\sin (\theta)}{\cos (\theta)+\sin (\theta)} $$

With the help of your classmates, determine the number of solutions to \(\sin (x)=\frac{1}{2}\) in \([0,2 \pi)\). Then find the number of solutions to \(\sin (2 x)=\frac{1}{2}, \sin (3 x)=\frac{1}{2}\) and \(\sin (4 x)=\frac{1}{2}\) in \([0,2 \pi)\). A pattern should emerge. Explain how this pattern would help you solve equations like \(\sin (11 x)=\frac{1}{2} .\) Now consider \(\sin \left(\frac{x}{2}\right)=\frac{1}{2}, \sin \left(\frac{3 x}{2}\right)=\frac{1}{2}\) and \(\sin \left(\frac{5 x}{2}\right)=\frac{1}{2} .\) What do you find? Replace \(\frac{1}{2}\) with \(-1\) and repeat the whole exploration.

In Exercises \(107-118\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. $$ \operatorname{arcsec}\left(\sec \left(\frac{4 \pi}{3}\right)\right) $$

Verify the identity. Assume that all quantities are defined. $$ \tan (\theta)+\cot (\theta)=\sec (\theta) \csc (\theta) $$

Verify the identity. Assume that all quantities are defined. $$ \csc (\theta)-\sin (\theta)=\cot (\theta) \cos (\theta) $$

In Exercises \(107-118\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. $$ \operatorname{arcsec}\left(\sec \left(\frac{5 \pi}{6}\right)\right) $$

Verify the identity. Assume that all quantities are defined. $$ \cos (\theta)-\sec (\theta)=-\tan (\theta) \sin (\theta) $$

In Exercises \(107-118\), assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. $$ \operatorname{arcsec}\left(\sec \left(\frac{5 \pi}{3}\right)\right) $$