91Ó°ÊÓ

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$2 \tan t-\sec ^{2} t=0$$

Use an addition or subtraction formula to find the solutions of the equation that are in the interval \([0, \pi)\). $$\cos 5 t \cos 2 t=-\sin 5 t \sin 2 t$$

Either show that the equation \(i s\) an identity or show that the equation is not an identity. $$\cos x(\tan x+\cot x)=\csc x$$

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$\tan \theta+\sec \theta=1$$

Use an addition or subtraction formula to find the solutions of the equation that are in the interval \([0, \pi)\). $$\sin 3 t \cos t+\cos 3 t \sin t=-\frac{1}{2}$$

Either show that the equation \(i s\) an identity or show that the equation is not an identity. $$\csc ^{2} x+\sec ^{2} x=\csc ^{2} x \sec ^{2} x$$

Use inverse trigonometric functions to find the solutions of the equation that are in the given interval, and approximate the solutions to four decimal places. $$3 \sin ^{2} t+7 \sin t+3=0 ; \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$

By definition, the average value of \(f(t)=c+a\) cos \(b t\) for one or more complete cycles is \(c\) (see the figure). (a) Use a double-angle formula to find the average value of \(f(t)=\sin ^{2} \omega t\) for \(0 \leq t \leq 2 \pi / \omega,\) with \(t\) in seconds. (b) In an electrical circuit with an alternating current \(I=I_{0} \sin \omega t,\) the rate \(r\) (in calories/sec) at which heat is produced in an \(R\) -ohm resistor is given by \(r=R I^{2}\) Find the average rate at which heat is produced for one complete cycle. (PICTURE CANNOT COPY)

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$\cot \alpha+\tan \alpha=\csc \alpha \sec \alpha$$

Make the trigonometric substitution \(x=a \sin \theta\) for \(-\pi / 2<\theta<\pi / 2\) and \(a>0 .\) Use fundamental identities to simplify the resulting expression. $$\frac{x^{2}}{a^{2}-x^{2}}$$