Test Objectives
• Demonstrate the ability to solve basic trigonometric equations
• Demonstrate the ability to solve trigonometric equations with half-angles
Solving Trigonometric Equations Part IV Practice Test:

#1:

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

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

$$b)\hspace{.1em}0=\text{cos}\hspace{.1em}\frac{θ}{2}+ \text{sin}\hspace{.1em}θ$$

$$c)\hspace{.1em}{-}1 + \text{sin}\hspace{.1em}\frac{θ}{2}+ \text{cos}\hspace{.1em}θ=0$$

#2:

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

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

$$b)\hspace{.1em}\hspace{.1em}\sqrt{3}\text{cos}\hspace{.1em}\frac{θ}{2}+ \text{cos}\hspace{.1em}θ + 1=0$$

$$c)\hspace{.1em}\hspace{.1em}\text{cos}\hspace{.1em}θ + 1=2\text{cos}\hspace{.1em}\frac{θ}{2}$$

#3:

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

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

$$b)\hspace{.1em}0=\text{cos}\hspace{.1em}θ + \text{cos}\hspace{.1em}\frac{θ}{2}$$

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

#4:

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

$$a)\hspace{.1em}{-}2 - \text{cos}\hspace{.1em}θ + 3 \text{cos}\hspace{.1em}\frac{θ}{2}=0$$

$$b)\hspace{.1em}2\sqrt{3}\text{tan}\frac{θ}{2}- 4=\sqrt{3}\text{tan}\frac{θ}{2}- 5$$

$$c)\hspace{.1em}2\text{tan}\frac{θ}{4}+ \sqrt{3}=2\sqrt{3}+ 3\text{tan}\frac{θ}{4}$$

#5:

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

$$a)\hspace{.1em}2\text{tan}\frac{θ}{2}- 5\text{sin}\hspace{.1em}θ=\text{tan}\frac{θ}{2}- 4\text{sin}\hspace{.1em}θ$$

$$b)\hspace{.1em}\sqrt{2}\text{tan}\frac{θ}{2}- \frac{\text{tan}\frac{θ}{2}}{\sqrt{2}}=\frac{\text{csc}\hspace{.1em}θ}{\sqrt{2}}$$

$$c)\hspace{.1em}\text{cos}\hspace{.1em}θ + 4\sqrt{2}\text{sin}\frac{θ}{2}=5\sqrt{2}\text{sin}\frac{θ}{2}+ \text{sin}\hspace{.1em}θ$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\left\{0\right\}$$ $$\left\{2πn, \frac{7π}{3}+ 4πn, \frac{11π}{3}+ 4πn\right\}$$

$$b)\hspace{.1em}\left\{π\right\}$$ $$\left\{π + 2πn, \frac{7π}{3}+ 4πn, \frac{11π}{3}+ 4πn\right\}$$

$$c)\hspace{.1em}\left\{0, \frac{π}{3}, \frac{5π}{3}\right\}$$ $$\left\{2πn, \frac{π}{3}+ 4πn, \frac{5π}{3}+ 4πn\right\}$$

#2:

Solutions:

$$a)\hspace{.1em}\left\{0, \frac{π}{3}\right\}$$ $$\left\{2πn, \frac{π}{3}+ 4πn, \frac{11π}{3}+ 4πn\right\}$$

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

$$c)\hspace{.1em}\left\{0, π\right\}$$ $$\left\{π + 2πn, 4πn\right\}$$

#3:

Solutions:

$$a)\hspace{.1em}\left\{0\right\}$$ $$\left\{2πn, \frac{8π}{3}+ 4πn, \frac{10π}{3}+ 4πn\right\}$$

$$b)\hspace{.1em}\left\{\frac{2π}{3}\right\}$$ $$\left\{\frac{2π}{3}+ \frac{4π}{3}n\right\}$$

$$c)\hspace{.1em}\left\{\frac{2π}{3}, π\right\}$$ $$\left\{π + 2πn, \frac{2π}{3}+ 4πn, \frac{10π}{3}+ 4πn\right\}$$

#4:

Solutions:

$$a)\hspace{.1em}\left\{0, \frac{2π}{3}\right\}$$ $$\left\{4πn, \frac{2π}{3}+ 4πn, \frac{10π}{3}+ 4πn\right\}$$

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

$$c)\hspace{.1em}\text{No Solution}$$ $$\left\{\frac{8π}{3}+ 4πn\right\}$$

#5:

Solutions:

$$a)\hspace{.1em}\left\{0, \frac{π}{2}, \frac{3π}{2}\right\}$$ $$\left\{\frac{π}{2}+ πn, 2πn\right\}$$

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

$$c)\hspace{.1em}\left\{\frac{π}{6}, \frac{3π}{2}\right\}$$ $$\left\{\frac{π}{6}+ 4πn, \frac{3π}{2}+ 4πn, \frac{5π}{2}+ 4πn, \frac{17π}{6}+ 4πn \right\}$$