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Section 3.2 Solving Trigonometric Equations
T: Be able to solve for an unknown angle and interpret the result in the appropriate quadrant.
Problem 3.2.1 .
Let
\(f(\theta)=2\cos(\theta)\text{.}\)
Use Desmos to solve
\(f(\theta)=-\sqrt{3}\) on the interval
\([0, 2\pi)\text{.}\) Sketch a graph below showing your solution(s).
Use the unit circle to solve
\(f(\theta)=-\sqrt{3}\) on the interval
\([0, 2\pi)\text{.}\) Sketch the unit circle below, showing your solutions(s).
Use inverse functions to solve
\(f(\theta)=-\sqrt{3}\) on the interval
\([0, 2\pi)\text{.}\) How can you be sure you have all the solutions between 0 and
\(2\pi\text{?}\)
Write an expression or expressions that represents all possible values of
\(\theta\) that will make the equation
\(f(\theta)=-\sqrt{3}\) true.
Problem 3.2.2 .
Solve each equation below on the interval
\([0, 2\pi)\text{.}\) Check your answers using Desmos.
\(\displaystyle 5+\cos\theta=6\)
\(\displaystyle 3\sqrt{2}=-6\sin\theta\)
\(\displaystyle -\frac{1}{2}=-\frac{1}{4}\sec\theta\)
\(\displaystyle -1=-1+4\tan\theta\)
\(\displaystyle 1+4\csc\theta=9\)
\(\displaystyle -4=-4+2\cot\theta\)
Problem 3.2.3 .
Let
\(g(\theta)=sin(2\theta)\text{.}\)
Use Desmos to solve
\(g(\theta)=\frac{1}{2}\) on the interval
\([0, 2\pi)\text{.}\) Sketch a graph below showing your solution(s). How many solutions do you have between 0 and
\(2\pi\text{?}\) Why?
Use the unit circle to solve
\(g(\theta)=\frac{1}{2}\) on the interval
\([0, 2\pi)\text{.}\) Sketch the unit circle below, showing your solutions(s).
Use inverse functions to solve
\(g(\theta)=\frac{1}{2}\) on the interval
\([0, 2\pi)\text{.}\) How can you be sure you have all the solutions between 0 and
\(2\pi\text{?}\)
Write an expression or expressions that represents all possible values of
\(\theta\) that will make the equation
\(g(\theta)=\frac{1}{2}\) true.
Problem 3.2.4 .
Solve each equation below on the interval
\([0, 2\pi)\text{.}\) Check your answers using Desmos.
\(\displaystyle \frac{\sqrt{3}}{2}=\cos(2\theta)\)
\(\displaystyle 1=\tan(\theta+\pi)\)
\(\displaystyle \sec(-\theta)=-1\)
\(\displaystyle \frac{1}{2}=\cos\!\left(\frac{\theta}{2}\right)\)
\(\displaystyle 3\cot\!\left(\theta+\frac{\pi}{3}\right)=3\sqrt{3}\)
\(\displaystyle \frac{5}{2}=3+\sin(-2\theta)\)
Problem 3.2.5 .
Solve each equation below on the interval
\([0, 2\pi)\text{.}\) Check your answers using Desmos.
\(\displaystyle \cos^2\theta-\cos\theta=0\)
\(\displaystyle 4\sin\theta-1=4\sin^2\theta\)
\(\displaystyle 2\tan\theta\sin\theta+\sqrt{3}\tan\theta=0\)
\(\displaystyle -2=2+\sec^2\theta+4\sec\theta\)
