91Ó°ÊÓ

In Exercises \(19-42,\) solve the equation, giving the exact solutions which lie in \([0,2 \pi)\) $$ \tan ^{3}(x)=3 \tan (x) $$

The Giant Wheel at Cedar Point is a circle with diameter 128 feet which sits on an 8 foot tall platform making its overall height is 136 feet. (Remember this from Exercise 17 in Section \(7.2 ?\) ) It completes two revolutions in 2 minutes and 7 seconds. \(^{20}\) Assuming the riders are at the edge of the circle, how fast are they traveling in miles per hour?

When I stand 30 feet away from a tree at home, the angle of elevation to the top of the tree is \(50^{\circ}\) and the angle of depression to the base of the tree is \(10^{\circ} .\) What is the height of the tree? Round your answer to the nearest foot.

In Exercises \(69-80,\) solve the inequality. Express the exact answer in interval notation, restricting your attention to \(0 \leq x \leq 2 \pi\). $$ \cot (x) \leq 4 $$

Find the exact value or state that it is undefined. $$ \arccos \left(\cos \left(\frac{2 \pi}{3}\right)\right) $$

Find the exact value or state that it is undefined. $$ \arccos \left(\cos \left(\frac{3 \pi}{2}\right)\right) $$

Find the exact value or state that it is undefined. $$ \arctan \left(\tan \left(\frac{\pi}{3}\right)\right) $$

In Exercises \(82-128\), verify the identity. Assume that all quantities are defined. $$ \frac{\cos (\theta)}{1+\sin (\theta)}=\frac{1-\sin (\theta)}{\cos (\theta)} $$

find the exact value or state that it is undefined. $$ \sin \left(\arccos \left(-\frac{1}{2}\right)\right) $$

As we did in Exercise 74 in Section \(10.2,\) let \(\alpha\) and \(\beta\) be the two acute angles of a right triangle. (Thus \(\alpha\) and \(\beta\) are complementary angles.) Show that \(\sec (\alpha)=\csc (\beta)\) and \(\tan (a)=\cot (\beta)\). The fact that co-fumetions of complementary angles are equal in this case is not an accident. and a more general result will be given in Section 10.4 .