Test Objectives
• Demonstrate the ability to evaluate trigonometric function composition
• Demonstrate the ability to solve trigonometric equations using inverses
• Demonstrate the ability to solve trigonometric equations involving inverse trigonometric functions
Solving Trigonometric Equations Involving Inverses Practice Test:

#1:

Instructions: Find the exact value.

$$a)\hspace{.1em}\text{tan}^{-1}\left(\text{tan}\left(\frac{7π}{3}\right)\right)$$

$$b)\hspace{.1em}\text{sin}^{-1}\left(\text{sin}\left(-\frac{17π}{6}\right)\right)$$

$$c)\hspace{.1em}\text{cos}^{-1}\left(\text{cos}\left(-\frac{7π}{5}\right)\right)$$

#2:

Instructions: Solve each equation for 0 ≤ x < 2π.

Try solving these equations using inverses.

$$a)\hspace{.1em}{-}2\text{csc}\left(\frac{x}{3}+ \frac{3π}{4}\right)=-4$$

$$b)\hspace{.1em}4\text{cos}\left(\frac{x}{2}+ \frac{π}{4}\right)=-2\sqrt{3}$$

$$c)\hspace{.1em}6\text{csc}\left(3x + \frac{5π}{6}\right)=-4\sqrt{3}$$

#3:

Instructions: Solve each equation for exact solutions.

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

$$b)\hspace{.1em}\text{sin}^{-1}\left(x - \frac{π}{3}\right)=\frac{π}{6}$$

$$c)\hspace{.1em}\frac{4}{7}\text{cos}^{-1}\left(\frac{x}{4}\right)=π$$ Hint: Consider the range of arccos, check all solutions.

#4:

Instructions: Solve each equation for exact solutions.

$$a)\hspace{.1em}\text{cos}^{-1}\frac{5}{13}=\text{tan}^{-1}(x)$$

$$b)\hspace{.1em}\text{sin}^{-1}\frac{8}{17}=\text{cos}^{-1}(x)$$

$$c)\hspace{.1em}\text{sec}^{-1}\frac{13}{12}=\text{cot}^{-1}(x)$$

#5:

Instructions: Solve each equation for exact solutions.

These identities will speed up your work. Try solving both ways.

$$\text{cos}^{-1}(x) + \text{sin}^{-1}(x)=\frac{π}{2}$$

$$\text{tan}^{-1}(x)=\text{sin}^{-1}\left(\frac{x}{\sqrt{x^2 + 1}}\right)$$

$$a)\hspace{.1em}\text{cos}^{-1}(x) - \text{sin}^{-1}(x)=\frac{7π}{6}$$

$$b)\hspace{.1em}\text{cos}^{-1}(x) + \text{tan}^{-1}(x)=\frac{π}{2}$$

$$c)\hspace{.1em}2\text{sin}^{-1}(x) + \text{cos}^{-1}(x)=π$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\frac{π}{3}$$

$$b)\hspace{.1em}{-}\frac{π}{6}$$

$$c)\hspace{.1em}\frac{3π}{5}$$

#2:

Solutions:

$$a)\hspace{.1em}x=\frac{π}{4}$$

$$b)\hspace{.1em}x=\frac{7π}{6}, \frac{11π}{6}$$

$$c)\hspace{.1em}x=\frac{π}{6}, \frac{5π}{18}, \frac{5π}{6}, \frac{17π}{18}, \frac{3π}{2}, \frac{29π}{18}$$

#3:

Solutions:

$$a)\hspace{.1em}x=\frac{\sqrt{2}}{2}$$

$$b)\hspace{.1em}x=\frac{1}{2}+ \frac{π}{3}$$

$$c)\hspace{.1em}\text{No Solution}$$

#4:

Solutions:

$$a)\hspace{.1em}x=\frac{12}{5}$$

$$b)\hspace{.1em}x=\frac{15}{17}$$

$$c)\hspace{.1em}x=\frac{12}{5}$$

#5:

Solutions:

$$a)\hspace{.1em}x=-\frac{\sqrt{3}}{2}$$

$$b)\hspace{.1em}x=0$$

$$c)\hspace{.1em}x=1$$