Chapter 13

In \(11-18,\) find all radian measures of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) that make each equation true. Express your answers in terms of \(\pi\) when possible; otherwise, to the nearest hundredth. $$ 2 \sin 2 \theta+\sin \theta=0 $$

In \(9-14,\) find, to the nearest tenth of a degree, the values of \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that satisfy each equation. $$ 25 \cos ^{2} \theta-4=0 $$

In \(9-14,\) find the exact values for \(\theta\) in the interval \(0 \leq \theta<2 \pi\) $$ \sin \theta+\sqrt{2}=\frac{\sqrt{2}}{2} $$

In \(3-14,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta < 360^{\circ}\) that satisfy each equation. $$ \cot ^{2} \theta=\csc \theta+1 $$

In \(11-18,\) find all radian measures of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) that make each equation true. Express your answers in terms of \(\pi\) when possible; otherwise, to the nearest hundredth. $$ 5 \sin ^{2} \theta-4 \sin \theta+\cos 2 \theta=0 $$

In \(9-14,\) find, to the nearest tenth of a degree, the values of \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that satisfy each equation. $$ \tan ^{2} \theta+4 \tan \theta-12=0 $$

In \(9-14,\) find the exact values for \(\theta\) in the interval \(0 \leq \theta<2 \pi\) $$ 3 \csc \theta+5=\csc \theta+9 $$

In \(3-14\) , use the quadratic formula to find, to the nearest degree, all values of \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that satisfy each equation. $$ \frac{\sin \theta}{2}=\frac{3}{\sin \theta+2} $$

In \(3-14,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta < 360^{\circ}\) that satisfy each equation. $$ 2 \sin ^{2} \theta-\tan \theta \cot \theta=0 $$

In \(11-18,\) find all radian measures of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) that make each equation true. Express your answers in terms of \(\pi\) when possible; otherwise, to the nearest hundredth. $$ \cos \theta=3 \sin 2 \theta $$

In \(9-14,\) find, to the nearest tenth of a degree, the values of \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that satisfy each equation. $$ \sec ^{2} \theta-7 \sec \theta+12=0 $$

In \(9-14,\) find the exact values for \(\theta\) in the interval \(0 \leq \theta<2 \pi\) $$ 4(\cot \theta+1)=2(\cot \theta+2) $$

Find all radian values of \(\theta\) in the interval \(0 \leq \theta<2 \pi\) for which \(\frac{\sin \theta}{1}=\frac{1}{2 \sin \theta}\)

An engineer would like to model a piece for a factory machine on his computer. As shown in the figure, the machine consists of a link fixed to a circle at point \(A\) . The other end of the link is fixed to a slider at point B. As the circle rotates, point \(B\) slides back and forth between the two ends of the slider \((C \text { and } D)\) . The movement is restricted so that \(\theta\) , the measure of \(\angle A O D,\) is in the interval \(-45^{\circ} \leq \theta \leq 45^{\circ} .\) The motion of point \(B\) can be described mathematically by the formula $$ C B=r(\cos \theta-1)+\sqrt{l^{2}-r^{2} \sin ^{2} \theta} $$ where \(r\) is the radius of the circle and \(l\) is the length of the link. Both the radius of the circle and the length of the link are 2 inches. a. Find the exact value of \(C B\) when: \((1) \theta=30^{\circ}(2) \theta=45^{\circ} .\) b. Find the exact value(s) of \(\theta\) when \(C B=2\) inches. c. Find, to the nearest hundredth of a degree, the value(s) of \(\theta\) when \(C B=1.5\) inches.

In \(11-18,\) find all radian measures of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) that make each equation true. Express your answers in terms of \(\pi\) when possible; otherwise, to the nearest hundredth. $$ 5 \sin ^{2} \theta-4 \sin \theta+\cos 2 \theta=0 $$

In \(15-20\) , find, to the nearest hundredth of a radian, the values of \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that satisfy the equation. $$ \tan ^{2} \theta-5 \tan \theta+6=0 $$

In \(15-20,\) find, to the nearest degree, the measure of an acute angle for which the given equation is true. $$ \sin \theta+3=5 \sin \theta $$

Find, to the nearest hundredth of a radian, all values of \(\theta\) in the interval \(0 \leq \theta<2 \pi\) for which \(\frac{\cos \theta}{3}=\frac{1}{3 \cos \theta+1}\)

In \(11-18,\) find all radian measures of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) that make each equation true. Express your answers in terms of \(\pi\) when possible; otherwise, to the nearest hundredth. $$ \sin \left(90^{\circ}-\theta\right)+\cos ^{2} \theta=\frac{1}{4} $$

In \(15-20\) , find, to the nearest hundredth of a radian, the values of \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that satisfy the equation. $$ 4 \cos ^{2} \theta-3 \cos \theta=1 $$

In \(15-20,\) find, to the nearest degree, the measure of an acute angle for which the given equation is true. $$ 3 \tan \theta-1=\tan \theta+9 $$

In \(11-18,\) find all radian measures of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) that make each equation true. Express your answers in terms of \(\pi\) when possible; otherwise, to the nearest hundredth. $$ 2 \cos ^{2} \theta+3 \sin \theta-2 \cos 2 \theta=1 $$

In \(15-20\) , find, to the nearest hundredth of a radian, the values of \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that satisfy the equation. $$ 5 \sin ^{2} \theta+2 \sin \theta=0 $$

In \(15-20,\) find, to the nearest degree, the measure of an acute angle for which the given equation is true. $$ 5 \cos \theta+1=8 \cos \theta $$

In \(11-18,\) find all radian measures of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) that make each equation true. Express your answers in terms of \(\pi\) when possible; otherwise, to the nearest hundredth. $$ (2 \sin \theta \cos \theta)^{2}+4 \sin 2 \theta-1=0 $$

In \(15-20\) , find, to the nearest hundredth of a radian, the values of \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that satisfy the equation. $$ 3 \sin ^{2} \theta+7 \sin \theta+2=0 $$

In \(15-20,\) find, to the nearest degree, the measure of an acute angle for which the given equation is true. $$ 4(\sin \theta+1)=6-\sin \theta $$

Martha swims 90 meters from point \(A\) on the north bank of a stream to point \(B\) on the opposite bank. Then she makes a right angle turn and swims 60 meters from point \(B\) to point \(C,\) another point on the north bank. If \(\mathrm{m} \angle C A B=\theta\) , then \(\mathrm{m} \angle A C B=90^{\circ}-\theta\) a. Let \(d\) be the width of the stream, the length of the perpendicular distance from \(B\) to \(\overline{A C}\) . Express \(d\) in terms of \(\sin \theta\) . b. Express \(d\) in terms of \(\sin \left(90^{\circ}-\theta\right)\) c. Use the answers to a and b to write an equation. Solve the equation for \(\theta .\) d. Find \(d,\) the width of the stream.

In \(15-20\) , find, to the nearest hundredth of a radian, the values of \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that satisfy the equation. $$ \csc ^{2} \theta-6 \csc \theta+8=0 $$

In \(15-20,\) find, to the nearest degree, the measure of an acute angle for which the given equation is true. $$ \csc \theta-1=3 \csc \theta-11 $$

In \(15-20\) , find, to the nearest hundredth of a radian, the values of \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that satisfy the equation. $$ 2 \cot ^{2} \theta-13 \cot \theta+6=0 $$

In \(15-20,\) find, to the nearest degree, the measure of an acute angle for which the given equation is true. $$ \cot \theta+8=3 \cot \theta+2 $$

Find the smallest positive value of \(\theta\) such that \(4 \sin ^{2} \theta-1=0\)

In \(21-24,\) find, to the nearest tenth, the degree measures of all \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that make the equation true. $$ 8 \cos \theta=3-4 \cos \theta $$

Find, to the nearest hundredth of a radian, the value of \(\theta\) such that \(\sec \theta=\frac{5}{\sec \theta}\) and \(\frac{\pi}{2} < \theta < \pi\)

In \(21-24,\) find, to the nearest tenth, the degree measures of all \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that make the equation true. $$ 5 \sin \theta-1=1-2 \sin \theta $$

Find two values of \(A\) such that \((\sin A)(\csc A)=-\sin A\)

In \(21-24,\) find, to the nearest tenth, the degree measures of all \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that make the equation true. $$ \tan \theta-4=3 \tan \theta+4 $$

In \(21-24,\) find, to the nearest tenth, the degree measures of all \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that make the equation true. $$ 2-\sec \theta=5+\sec \theta $$

In \(25-28,\) find, to the nearest hundredth, the radian measures of all \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that make the equation true. $$ 10 \sin \theta+1=3-2 \sin \theta $$

In \(25-28,\) find, to the nearest hundredth, the radian measures of all \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that make the equation true. $$ 9-2 \cos \theta=8-4 \cos \theta $$

In \(25-28,\) find, to the nearest hundredth, the radian measures of all \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that make the equation true. $$ 15 \tan \theta-7=5 \tan \theta-3 $$

In \(25-28,\) find, to the nearest hundredth, the radian measures of all \(\theta\) in the interval \(0 \leq \theta<2 \pi\) that make the equation true. $$ \cot \theta-6=2 \cot \theta+2 $$

The voltage \(E\) (in volts) in an electrical circuit is given by the function $$ E=20 \cos (\pi t) $$ where \(t\) is time in seconds. a. Graph the voltage \(E\) in the interval \(0 \leq t \leq 2\) . b. What is the voltage of the electrical circuit when \(t=1 ?\) c. How many times does the voltage equal 12 volts in the first two seconds? d. Find, to the nearest hundredth of a second, the times in the first two seconds when the voltage is equal to 12 volts. (1) Let \(\theta=\pi t .\) Solve the equation \(20 \cos \theta=12\) in the interval \(0 \leq \theta<2 \pi\) (2) Use the formula \(\theta=\pi t\) and your answers to part \((1)\) to find \(t\) when \(0 \leq \theta<2 \pi\) and the voltage is equal to 12 volts.

A water balloon leaves the air cannon at an angle of \(\theta\) with the ground and an initial velocity of 40 feet per second. The water balloon lands 30 feet from the cannon. The distance \(d\) traveled by the water balloon is given by the formula $$ d=\frac{1}{32} v^{2} \sin 2 \theta $$ where \(v\) is the initial velocity. a. Let \(x=2 \theta .\) Solve the equation \(30=\frac{1}{32}(40)^{2} \sin x\) to the nearest tenth of a degree. b. Use the formula \(x=2 \theta\) and your answer to part a to find the measure of the angle that the cannon makes with the ground.

It is important to understand the underlying mathematics before using the calculator to solve trigonometric equations. For example, Adrian tried to use the intersect feature of his graphing calculator to find the solutions of the equation cot \(\theta=\sin \left(\theta-\frac{\pi}{2}\right)\) in the interval \(0 \leq \theta \leq \pi\) but got an error message. Follow the steps that Adrian used to solve the equation: (1) Enter \(Y_{1}=\frac{1}{\tan X}\) and \(Y_{2}=\sin \left(X-\frac{\pi}{2}\right)\) into the \(Y=\) menu. (2) Use the following viewing window to graph the equations: $$ X \min =0, \operatorname{Xmax}=\pi, X s c l=\frac{\pi}{6}, Y \min =-5, Y \max =5 $$ (3) The curves seem to intersect at \(\left(\frac{\pi}{2}, 0\right) .\) Press 2nd CALC 5 ENTER ENTER to select both curves. When the calculator asks for a guess, move the cursor near the intersection point using the arrow keys and then press ENTER a. Why does the calculator return an error message? b. Is \(\theta=\frac{\pi}{2}\) a solution to the equation? Explain.