91Ó°ÊÓ

Sketch the graph of the trigonometric function \(y=\sin \theta .\) On the same axes, sketch the graph of \(y=\csc \theta\) using the fact that the \(y\) -value for \(\csc \theta\) is the reciprocal of the corresponding \(y\) -value for \(\sin \theta .\) Where do the asymptotes occur in the graph of the cosecant function?

Calculate the exact value of the inverse function geometrically. Assume the principal branch in all cases. Check your answers by direct calculation. $$\sec \left(\sin ^{-1} \frac{15}{17}\right)$$

Show the steps in trans forming the expression on the left to the one on the right. $$\begin{aligned} &\cot D \cos D+\sin D \quad \text { to }\\\ &\csc D \end{aligned}$$

On your grapher, make a table with columns showing the values of the trigonometric expressions \(\tan ^{2} \theta\) and \(\sec ^{2} \theta\) for \(0^{\circ}, 15^{\circ}\) \(30^{\circ}, \ldots .\) What relationship do you notice between the two columns? How do you explain this relationship? How do you explain what happens at \(90^{\circ} ?\)

Show algebraically that \(\sin ^{2} x=1-\cos ^{2} x\)

Calculate the exact value of the inverse function geometrically. Assume the principal branch in all cases. Check your answers by direct calculation. $$\cos \left(\sin ^{-1}\left(-\frac{8}{17}\right)\right)$$

Show algebraically that \(\cot ^{2} x=\csc ^{2} x-1\)

Show the steps in trans forming the expression on the left to the one on the right. $$\begin{aligned} &\sec x-\cos x \quad \text { to }\\\ &\sin x \tan x \end{aligned}$$

Calculate the exact value of the inverse function geometrically. Assume the principal branch in all cases. Check your answers by direct calculation. $$\sec \left(\cos ^{-1} \frac{2}{3}\right)(\text { Surprise } ?)$$

Show the steps in trans forming the expression on the left to the one on the right. \(\tan x(\cot x \cos x+\sin x)\) to sec \(x\)