91Ó°ÊÓ

Use the product-to-sum identities and the sum-to-product identities to prove each of the following. $$\tan \frac{\theta+\phi}{2} \tan \frac{\phi-\theta}{2}=\frac{\cos \theta-\cos \phi}{\cos \theta+\cos \phi}$$

Evaluate. \(\cos ^{-1}(\sin \pi)\)

Consider the following functions ( \(a\) )- ( \(f\) ). Without graphing them, answer question. a) \(f(x)=2 \sin \left(\frac{1}{2} x-\frac{\pi}{2}\right)\) b) \(f(x)=\frac{1}{2} \cos \left(2 x-\frac{\pi}{4}\right)+2\) c) \(f(x)=-\sin \left[2\left(x-\frac{\pi}{2}\right)\right]+2\) d \(f(x)=\sin (x+\pi)-\frac{1}{2}\) e) \(f(x)=-2 \cos (4 x-\pi)\) f) \((x)=-\cos \left[2\left(x-\frac{\pi}{8}\right)\right]\) Which functions have a graph with a period of \(\pi ?\)

Solve using a calculator, finding all solutions in \([0,2 \pi)\). $$\cos x-2=x^{2}-3 x$$

Use the product-to-sum identities and the sum-to-product identities to prove each of the following. $$\begin{aligned}\tan \frac{\theta+\phi}{2}(\sin \theta-\sin \phi) &=& \tan \frac{\theta-\phi}{2}(\sin \theta+\sin \phi)\end{aligned}$$

Consider the following functions ( \(a\) )- ( \(f\) ). Without graphing them, answer question. a) \(f(x)=2 \sin \left(\frac{1}{2} x-\frac{\pi}{2}\right)\) b) \(f(x)=\frac{1}{2} \cos \left(2 x-\frac{\pi}{4}\right)+2\) c) \(f(x)=-\sin \left[2\left(x-\frac{\pi}{2}\right)\right]+2\) d \(f(x)=\sin (x+\pi)-\frac{1}{2}\) e) \(f(x)=-2 \cos (4 x-\pi)\) f) \((x)=-\cos \left[2\left(x-\frac{\pi}{8}\right)\right]\) Which functions have a graph with a period of \(2 \pi ?\)

Evaluate. \(\sin ^{-1}\left[\tan \left(-\frac{\pi}{4}\right)\right]\)

Use the product-to-sum identities and the sum-to-product identities to prove each of the following. $$\sin 2 \theta+\sin 4 \theta+\sin 6 \theta=4 \cos \theta \cos 2 \theta \sin 3 \theta$$

Use the given substitution to express the given radical expression as a trigonometric function without radicals. Assume that \(0<\theta<\pi / 2\) Let \(x=4 \sec \theta\) in \(\frac{\sqrt{x^{2}-16}}{x^{2}}\)

Consider the following functions ( \(a\) )- ( \(f\) ). Without graphing them, answer question. a) \(f(x)=2 \sin \left(\frac{1}{2} x-\frac{\pi}{2}\right)\) b) \(f(x)=\frac{1}{2} \cos \left(2 x-\frac{\pi}{4}\right)+2\) c) \(f(x)=-\sin \left[2\left(x-\frac{\pi}{2}\right)\right]+2\) d \(f(x)=\sin (x+\pi)-\frac{1}{2}\) e) \(f(x)=-2 \cos (4 x-\pi)\) f) \((x)=-\cos \left[2\left(x-\frac{\pi}{8}\right)\right]\) Which functions have a graph with a phase shift of \(\frac{\pi}{4} ?\)