91Ó°ÊÓ

Find the exact value of each expression. $$\tan ^{2}\left(\frac{1}{2} \sin ^{-1} \frac{\sqrt{3}}{2}\right)$$

Find the exact value of each expression. $$\tan ^{2}\left(\frac{1}{2} \cos ^{-1} \frac{\sqrt{3}}{2}\right)$$

In Exercises \(83-88,\) find the exact value of each expression. $$\cos \left(\sin ^{-1} 0+\cos ^{-1} \frac{1}{2}\right)$$

In Exercises \(81-86,\) use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities. $$\sin 2 x=2 \sin x \cos x$$

Use a graphing utility to find the solutions of the given equations, in radians, that lie in the interval \([0,2 \pi)\). $$\sin 2 x=1-x$$

The horizontal range of a projectile fired with an initial velocity of 40 meters per second at an angle \(\theta\) is given by \(R=\frac{40^{2} \sin 2 \theta}{9.8} .\) Find \(R\) to four decimal places if it is known that \(\sin \theta=0.3\) and \(\theta\) is in the first quadrant.

Use a graphing utility to find the solutions of the given equations, in radians, that lie in the interval \([0,2 \pi)\). $$\tan x=x+2$$

In Exercises \(81-86,\) use a graphing utility to graph each side of the equation and decide whether the equation is an identity. You need not verify the ones that are identities. $$\sin (x-\pi)=\sin x$$

In Exercises \(83-88,\) find the exact value of each expression. $$\cos \left(\sin ^{-1} \frac{3}{5}+\frac{\pi}{2}\right)$$

The expression \(\sin (x+c t)+\sin (x-c t)\) represents a traveling wave that is moving at speed \(c .\) (a) Write the expression in terms of a product of functions. (b) \(^{4}\) The function \(f(x, t)=\sin (x-t)\) is a function of "two variables, \(x\) and \(t\), where \(x\) stands for position and t represents time. For a fixed value of \(t, \sin (x-t)\) is a function of \(x\) alone. For each of three fixed values of \(t(t=0, t=1, \text { and } t=2),\) graph this function. What happens to the graph as \(t\) increases?