Test Objectives
• Demonstrate the ability to solve a trigonometric equation with multiple angles
• Demonstrate the ability to solve a trigonometric equation with steps inside
• Demonstrate the ability to solve a trigonometric equation with identities
Solving Trigonometric Equations Part V Practice Test:

#1:

Instructions: Solve each for 0 ≤ θ < 2π and for all solutions.

$$a)\hspace{.1em}2\text{cos}\hspace{.1em}θ=\text{sin}\hspace{.1em}2θ$$

$$b)\hspace{.1em}2\text{cos}^2 θ - 2=-\text{cos}\hspace{.1em}2θ$$

$$c)\hspace{.1em}\text{cos}\hspace{.1em}2θ=-2\text{cos}^2 θ$$

#2:

Instructions: Solve each for 0 ≤ θ < 2π and for all solutions.

$$a)\hspace{.1em}2\text{sin}\hspace{.1em}2θ=-\sqrt{3}\text{sin}\hspace{.1em}θ + 3\text{sin}\hspace{.1em}2 θ$$

$$b)\hspace{.1em}4\text{cos}^2 θ - \text{sin}^2 2θ=0$$

$$c)\hspace{.1em}2\text{cos}^2 θ=1 - \text{cos}\hspace{.1em}2θ$$

#3:

Instructions: Solve each for 0 ≤ θ < 2π and for all solutions.

$$a)\hspace{.1em}{-}\frac{1}{5}\text{csc}\left(θ + \frac{π}{2}\right)=\frac{2}{5}$$

$$b)\hspace{.1em}{-}\text{cos}\left(3θ + \frac{3π}{2}\right)=-\frac{\sqrt{2}}{2}$$

$$c)\hspace{.1em}\sqrt{3}=2\text{cos}\left(\frac{θ}{4}+ \frac{5π}{3}\right)$$

#4:

Instructions: Solve each for all solutions.

$$a)\hspace{.1em}{-}\text{sin}\hspace{.1em}(x)-\text{cos}(4x)=-\text{cos}(6x)$$

$$b)\hspace{.1em}\text{sin}(2x) + \text{cos}(7x)=\text{cos}(3x)$$

$$c)\hspace{.1em}\text{cos}(4x) + \text{sin}(3x)=\text{sin}(5x)$$

#5:

Instructions: Solve each for all solutions.

$$a)\hspace{.1em}\text{cos}(2x)=\text{sin}(7x) + \text{sin}(3x)$$

$$b)\hspace{.1em}\text{cos}(5x) + \text{cos}(3x)=\text{cos}(x)$$

$$c)\hspace{.1em}{-}\text{cos}(3x)=\text{cos}(5x) + \text{cos}(x)$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\left\{\frac{π}{2}, \frac{3π}{2}\right\}$$ $$\left\{\frac{π}{2}+ πn\right\}$$

$$b)\hspace{.1em}\left\{\frac{π}{6}, \frac{5π}{6}, \frac{7π}{6}, \frac{11π}{6}\right\}$$ $$\left\{\frac{π}{6}+ πn, \frac{5π}{6}+ πn\right\}$$

$$c)\hspace{.1em}\left\{\frac{π}{3}, \frac{2π}{3}, \frac{4π}{3}, \frac{5π}{3}\right\}$$ $$\left\{\frac{π}{3}+ πn, \frac{2π}{3}+ πn\right\}$$

#2:

Solutions:

$$a)\hspace{.1em}\left\{0, \frac{π}{6}, π, \frac{11π}{6}\right\}$$ $$\left\{πn, \frac{π}{6}+ 2πn, \frac{11π}{6}+ 2πn\right\}$$

$$b)\hspace{.1em}\left\{\frac{π}{2}, \frac{3π}{2}\right\}$$ $$\left\{\frac{π}{2}+ πn\right\}$$

$$c)\hspace{.1em}\left\{\frac{π}{4}, \frac{3π}{4}, \frac{5π}{4}, \frac{7π}{4}\right\}$$ $$\left\{\frac{π}{4}+ \frac{π}{2}n\right\}$$

#3:

Solutions:

$$a)\hspace{.1em}\left\{\frac{2π}{3}, \frac{4π}{3}\right\}$$ $$\left\{\frac{2π}{3}+ 2πn, \frac{4π}{3}+ 2πn\right\}$$

$$b)\hspace{.1em}\left\{\frac{π}{12}, \frac{π}{4}, \frac{3π}{4}, \frac{11π}{12}, \frac{17π}{12},\frac{19π}{12}\right\}$$ $$\left\{\frac{π}{12}+ \frac{2π}{3}n, \frac{π}{4}+ \frac{2π}{3}n\right\}$$

$$c)\hspace{.1em}\left\{\frac{2π}{3}\right\}$$ $$\left\{\frac{2π}{3}+ 8πn, 2π + 8πn\right\}$$

#4:

Solutions:

$$a)\hspace{.1em}\left\{πn, \frac{7π}{30}+ \frac{2π}{5}n, \frac{11π}{30}+ \frac{2π}{5}n\right\}$$

$$b)\hspace{.1em}\left\{\frac{π}{2}n, \frac{π}{30}+ \frac{2π}{5}n, \frac{π}{6}+ \frac{2π}{5}n\right\}$$

$$c)\hspace{.1em}\left\{\frac{π}{8}+ \frac{π}{4}n, \frac{π}{6}+ 2πn, \frac{5π}{6}+ 2πn\right\}$$

#5:

Solutions:

$$a)\hspace{.1em}\left\{\frac{π}{4}+ \frac{π}{2}n, \frac{π}{30}+ \frac{2π}{5}n, \frac{π}{6}+ \frac{2π}{5}n\right\}$$

$$b)\hspace{.1em}\left\{\frac{π}{2}+ πn, \frac{π}{12}+ \frac{π}{2}n, \frac{5π}{12}+ \frac{π}{2}n\right\}$$

$$c)\hspace{.1em}\left\{\frac{π}{2}+ πn, \frac{π}{6}+ \frac{π}{2}n, \frac{π}{3}+ \frac{π}{2}n\right\}$$