91Ó°ÊÓ

In Exercises \(69-72\), find all solutions of the equation in the interval \([0,2 \pi)\). $$ \sin \left(x+\frac{\pi}{3}\right)+\sin \left(x-\frac{\pi}{3}\right)=1 $$

In Exercises 69-72, use a graphing utility to complete the table and graph the functions. Make a conjecture about \(y_{1}\) and \(y_{2}\). \(\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}\) $$ y_{1}=\cos \left(\frac{\pi}{2}-x\right), \quad y_{2}=\sin x $$

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 \(0 \leq t \leq 1\).

In Exercises 63-74, use the product-to-sum formulas to write the product as a sum or difference. $$ 5 \cos (-5 \beta) \cos 3 \beta $$

In Exercises 63-74, use the product-to-sum formulas to write the product as a sum or difference. $$ \cos 2 \theta \cos 4 \theta $$

The displacement from equilibrium of a weight oscillating on the end of a spring is given by \(y=1.56 t^{-1 / 2} \cos 1.9 t\), where \(y\) is the displacement (in feet) and \(t\) is the time (in seconds). Use a graphing utility to graph the displacement function for \(0 \leq t \leq 10\). Find the time beyond which the displacement does not exceed 1 foot from equilibrium.

In Exercises \(69-72\), find all solutions of the equation in the interval \([0,2 \pi)\). $$ \sin \left(x+\frac{\pi}{6}\right)-\sin \left(x-\frac{\pi}{6}\right)=\frac{1}{2} $$

In Exercises 69-72, use a graphing utility to complete the table and graph the functions. Make a conjecture about \(y_{1}\) and \(y_{2}\). \(\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}\) $$ y_{1}=\sec x-\cos x, \quad y_{2}=\sin x \tan x $$

In Exercises 69-72, use a graphing utility to complete the table and graph the functions. Make a conjecture about \(y_{1}\) and \(y_{2}\). \(\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline y_{1} & & & & & & & \\ \hline y_{2} & & & & & & & \\ \hline \end{array}\) $$ y_{1}=\frac{\cos x}{1-\sin x}, \quad y_{2}=\frac{1+\sin x}{\cos x} $$

In Exercises \(69-72\), find all solutions of the equation in the interval \([0,2 \pi)\). $$ \cos \left(x+\frac{\pi}{4}\right)-\cos \left(x-\frac{\pi}{4}\right)=1 $$