91Ó°ÊÓ

Verifying Solutions In Exercises \(5-10\) , verify that the \(x\) -values are solutions of the equation. $$\begin{array}{l}{\csc ^{4} x-4 \csc ^{2} x=0} \\ {\text { (a) } x=\frac{\pi}{6}}\end{array}$$ $$x=\frac{5 \pi}{6}$$

Using a Double-Angle Formula In Exercises \(15-20\) , use a double-angle formula to rewrite the expression. $$6 \sin x \cos x$$

Evaluating Trigonometric Functions In Exercises \(11-26,\) find the exact values of the sine, cosine, and tangent of the angle. $$ 15^{\circ} $$

Verify the identity. $$\frac{\cos \theta \cot \theta}{1-\sin \theta}-1=\csc \theta$$

Using Half-Angle Formulas, (a) determine the quadrant in which \(u\) 2 lies, and (b) find the exact values of \(\sin (u\) 2), \(\cos (u\) 2), and \(\tan (u\) 2) using the half-angle formulas. $$\tan u=-5 / 12, \quad 3 \pi / 2

Solving a Trigonometric Equation, find all solutions of the equation in the interval\(0,2 \pi\) ). Use a graphing utility to graph the equation and verify the solutions. $$\cos \frac{x}{2}-\sin x=0$$

Using Product-to-Sum Formulas, use the product-to-sum formulas to rewrite the product as a sum or difference. $$ \cos 2 \theta \cos 4 \theta $$

66\. Shadow Length. The length \(s\) of a shadow cast by a vertical gnomon (a device used to tell time) of height \(h\) when the angle of the sun above the horizon is \(\theta\) can be modeled by the equation $$s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}.$$ \(\begin{array}{l}{\text { (a) Verify that the expression for } s \text { is equal to } h \text { cot } \theta \text { . }} \\ {\text { (b) Use a graphing utility to complete the table. Let }} {h=5 \text { feet. }}\end{array}\) \(\begin{array}{l}{\text { (c) Use your table from part (b) to determine the }} \\\ {\text { angles of the sun that result in the maximum and }} \\ {\text { minimum lengths of the shadow. }} \\ {\text { (d) Based on your results from part (c), what time of }} \\ {\text { day do you think it is when the angle of the sun }} \\ {\text { above the horizon is } 90^{\circ} ?}\end{array}\)

Projectile Motion A baseball is hit at an angle of \(\theta\) with the horizontal and with an initial velocity of \(v_{0}=100\) feet per second. An outfielder catches the ball 300 feet from home plate (see figure). Find \(\theta\) when the range \(r\) of a projectile is given by $$r=\frac{1}{32} v_{0}^{2} \sin 2 \theta$$

Angle Between Two Lines In Exercises 97 and 98 , use the figure, which shows two lines whose equations, are \(y_{1}=m_{1} x+b_{1}\) and \(y_{2}=m_{2} x+b_{2} .\) Assume that both lines have positive slopes. Derive a formula for the angle between the two lines. Then use your formula to find the angle between the given pair of lines. \(y=x\) and \(y=\frac{1}{\sqrt{3}} x\)