91Ó°ÊÓ

Use the product-to-sum identities and the sum-to-product identities to prove each of the following. $$\begin{array}{c}\cot 4 x(\sin x+\sin 4 x+\sin 7 x) =\cos x+\cos 4 x+\cos 7 x\end{array}$$

Use the given substitution to express the given radical expression as a trigonometric function without radicals. Assume that \(a>0\) and \(0<\theta<\pi / 2 .\) Then find expressions for the indicated trigonometric functions. Let \(x=a \sin \theta\) in \(\sqrt{a^{2}-x^{2}} .\) Then find \(\cos \theta\) and \(\tan \theta\)

Evaluate. \(\cos \left(\sin ^{-1} \frac{\sqrt{3}}{2}\right)\)

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

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 an amplitude of \(2 ?\)

Evaluate. \(\tan \left(\cos ^{-1} \frac{\sqrt{2}}{2}\right)\)

Use the product-to-sum identities and the sum-to-product identities to prove each of the following. $$\cot \frac{x+y}{2}=\frac{\sin y-\sin x}{\cos x-\cos y}$$

Use the given substitution to express the given radical expression as a trigonometric function without radicals. Assume that \(a>0\) and \(0<\theta<\pi / 2 .\) Then find expressions for the indicated trigonometric functions. Let \(x=3 \sec \theta\) in \(\sqrt{x^{2}-9}\). Then find \(\sin \theta\) and \(\cos \theta\)

Solve using a calculator, finding all solutions in \([0,2 \pi)\). $$2 \cos ^{2} x=x+1$$

Use the given substitution to express the given radical expression as a trigonometric function without radicals. Assume that \(a>0\) and \(0<\theta<\pi / 2 .\) Then find expressions for the indicated trigonometric functions. Let \(x=a \sec \theta\) in \(\sqrt{x^{2}-a^{2}} .\) Then find \(\sin \theta\) and \(\cos \theta\)