Chapter 13

Sasha said that \(\sin \theta+\cos \theta=2\) has no solution. Do you agree with Sasha? Explain why or why not.

Isaiah said that if the equation \(\cos 2 x+2 \cos ^{2} x=2\) is divided by \(2,\) an equivalent equation is \(\cos x+\cos ^{2} x=1 .\) Do you agree with Isaiah? Explain why or why not.

The discriminant of the quadratic equation \(\tan ^{2} \theta+4 \tan \theta+5=0\) is \(-4 .\) Explain why the solution set of this equation is the empty set.

Can the equation \(\tan \theta+\sin \theta \tan \theta=1\) be solved by factoring the left side of the equation? Explain why or why not.

Explain why the solution set of the equation \(2 x+4=8\) is \(\\{2\\}\) but the solution set of the equation \(2 \sin x+4=8\) is \(\\{ \\},\) the empty set.

Explain why the solution set of \(2 \csc ^{2} \theta-\csc \theta=0\) is the empty set.

For what values of \(\theta\) is \(\sin \theta=\sqrt{1-\cos ^{2} \theta}\) true?

Aaron solved the equation \(2 \sin \theta \cos \theta=\cos \theta\) by first dividing both sides of the equation by cos \(\theta\) . Aaron said that for \(0 \leq \theta \leq 2 \pi,\) the solution set is \(\left\\{\frac{\pi}{6}, \frac{5 \pi}{6}\right\\} .\) Do you agree with Aaron? Explain why or why not.

Can the equation \(2(\sin \theta)(\cos \theta)+\sin \theta+2 \cos \theta+1=0\) be solved by factoring the left side of the equation? Explain why or why not.

Explain why \(2 x+4=8\) has only one solution in the set of real numbers but the equation \(2 \tan x+4=8\) has infinitely many solutions in the set of real numbers.

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. $$ 3 \sin ^{2} \theta-7 \sin \theta-3=0 $$

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

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ \sin 2 \theta-\cos \theta=0 $$

In \(3-8,\) find the exact solution set of each equation if \(0^{\circ} \leq \theta<360^{\circ} .\) $$ 2 \sin ^{2} \theta+\sin \theta-1=0 $$

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. $$ \tan ^{2} \theta-2 \tan \theta-5=0 $$

In \(3-14,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta < 360^{\circ}\) that satisfy each equation. $$ 4 \cos ^{2} \theta+4 \sin \theta-5=0 $$

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ \cos 2 \theta+\sin ^{2} \theta=1 $$

In \(3-8,\) find the exact solution set of each equation if \(0^{\circ} \leq \theta<360^{\circ} .\) $$ 3 \tan ^{2} \theta=1 $$

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. $$ 7 \cos ^{2} \theta-1=5 \cos \theta $$

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

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ \sin 2 \theta+2 \sin \theta=0 $$

In \(3-8,\) find the exact solution set of each equation if \(0^{\circ} \leq \theta<360^{\circ} .\) $$ \tan ^{2} \theta-3=0 $$

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. $$ 9 \sin ^{2} \theta+6 \sin \theta=2 $$

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

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ \tan 2 \theta=\cot \theta $$

In \(3-8,\) find the exact solution set of each equation if \(0^{\circ} \leq \theta<360^{\circ} .\) $$ 2 \sin ^{2} \theta-1=0 $$

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. $$ \tan ^{2} \theta+3 \tan \theta+1=0 $$

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

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ \cos 2 \theta+2 \cos ^{2} \theta=2 $$

In \(3-8,\) find the exact solution set of each equation if \(0^{\circ} \leq \theta<360^{\circ} .\) $$ 6 \cos ^{2} \theta+5 \cos \theta-4=0 $$

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. $$ 8 \cos ^{2} \theta-7 \cos \theta+1=0 $$

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

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ \sin \frac{1}{2} \theta=\cos \theta $$

In \(3-8,\) find the exact solution set of each equation if \(0^{\circ} \leq \theta<360^{\circ} .\) $$ 2 \sin \theta \cos \theta+\cos \theta=0 $$

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. $$ 2 \cot ^{2} \theta+3 \cot \theta-4=0 $$

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

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ 3-3 \sin \theta-2 \cos ^{2} \theta=0 $$

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. $$ \tan ^{2} \theta-3 \tan \theta+2=0 $$

In \(9-14,\) find the exact values for \(\theta\) in the interval \(0 \leq \theta<2 \pi\) $$ 3 \sin \theta-\sqrt{3}=\sin \theta $$

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

In \(3-10,\) find the exact values of \(\theta\) in the interval \(0^{\circ} \leq \theta \leq 360^{\circ}\) that make each equation true. $$ 3 \cos 2 \theta-4 \cos ^{2} \theta+2=0 $$

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. $$ 3 \cos ^{2} \theta-4 \cos \theta+1=0 $$

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

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. $$ 3 \csc ^{2} \theta-2 \csc \theta=2 $$

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

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. $$ \cos 2 \theta=2 \cos \theta-2 \cos ^{2} \theta $$

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. $$ 9 \sin ^{2} \theta-9 \sin \theta+2=0 $$

In \(9-14,\) find the exact values for \(\theta\) in the interval \(0 \leq \theta<2 \pi\) $$ \tan \theta+12=2 \tan \theta+11 $$

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. $$ 2 \tan \theta(\tan \theta+1)=3 $$

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