91Ó°ÊÓ

Find exact values for \(\sin \theta, \cos \theta,\) and \(\tan \theta\) using the information given. $$\sin (2 \theta)=-\frac{240}{289} ; 2 \theta \text { in QIII }$$

Use a cofunction identity to write an equivalent expression. \tan \left(\frac{5 \pi}{12}\right)

Rewrite in terms of an expression containing only cosines to the power \(1 .\) $$\sin ^{4} x \cos ^{2} x$$

Rewrite in terms of an expression containing only cosines to the power \(1 .\). $$3 \cos ^{4} x$$

Find all solutions in \([0,2 \pi)\). $$4 \sqrt{2} \sin ^{2} x=4 \sqrt{2}$$

Use sum/difference identities to verify that both expressions give the same result. a. \(\sin \left(\frac{\pi}{3}-\frac{\pi}{4}\right)\) b. \(\sin \left(\frac{\pi}{4}-\frac{\pi}{6}\right)\)

Use a half-angle identity to rewrite each expression as a single, nonradical function. $$\sqrt{\frac{1+\cos 30^{\circ}}{2}}$$

Simplify each expression without using a calculator. $$\sin ^{-1}\left[\cos \left(\frac{2 \pi}{3}\right)\right]$$

If two waves of the same frequency, velocity, and amplitude are traveling along a string in opposite directions, they can be represented by the equations \(Y_{1}=A \sin (k x-\omega t)\) and \(Y_{2}=A \sin (k x+\omega t) .\) Use the sum and difference formulas for sine to show the result \(\mathrm{Y}_{R}=\mathrm{Y}_{1}+\mathrm{Y}_{2}\) of these waves can be expressed as \(\mathrm{Y}_{R}=2 A \sin (k x) \cos (\omega t)\).

Working with identities: Compute the value of \(\cos 15^{\circ}\) two ways, first using the half-angle identity for cosine, and second using the difference identity for cosine. (a) Find a decimal approximation for each to show the results are equivalent and (b) verify algebraically that they are equivalent.