91Ó°ÊÓ

Use the product-to-sum formulas to write the product as a sum or difference. $$\sin 5 \theta \sin 3 \theta$$

Use the product-to-sum formulas to write the product as a sum or difference. $$3 \sin (-4 \alpha) \sin 6 \alpha$$

Use a graphing utility to complete the table and graph the functions. Make a conjecture about \(y_{1}\) and \(y_{2}\). $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array} $$ $$y_{1}=\sec x-\cos x, \quad y_{2}=\sin x \tan x$$

Use a graphing utility to approximate the solutions in the interval \([0,2 \pi)\). $$\tan (x+\pi)-\cos \left(x+\frac{\pi}{2}\right)=0$$

Use a graphing utility to complete the table and graph the functions. Make a conjecture about \(y_{1}\) and \(y_{2}\). $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array} $$ $$y_{1}=\frac{\cos x}{1-\sin x}, \quad y_{2}=\frac{1+\sin x}{\cos x}$$

Use a graphing utility to approximate the solutions in the interval \([0,2 \pi)\). $$\sin \left(x+\frac{\pi}{2}\right)+\cos ^{2} x=0$$

Use the product-to-sum formulas to write the product as a sum or difference. $$7 \cos (-5 \beta) \sin 3 \beta$$

Use the product-to-sum formulas to write the product as a sum or difference. $$\cos 2 \theta \cos 4 \theta$$

Use a graphing utility to approximate the solutions in the interval \([0,2 \pi)\). $$\cos \left(x-\frac{\pi}{2}\right)-\sin ^{2} x=0$$

A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the point of equilibrium is given by \(y=\frac{1}{12}(\cos 8 t-3 \sin 8 t),\) where \(y\) is the displacement (in meters) and \(t\) is the time (in seconds). Find the times when the weight is at the point of equilibrium \((y=0)\) for \(0 \leq t \leq 1\).