91Ó°ÊÓ

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 \(3-10,\) 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 the exact values for \(\theta\) in the interval \(0 \leq \theta<2 \pi\) $$ 4(\cot \theta+1)=2(\cot \theta+2) $$

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

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.