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

#1:

Instructions: Find the exact value.

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

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


#2:

Instructions: Solve each equation for exact solutions.

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

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


#3:

Instructions: Solve each equation for exact solutions.

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

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


#4:

Instructions: Solve each equation for exact solutions.

$$a)\hspace{.1em}\text{sin}^{-1}\frac{3}{5}=\text{cos}^{-1}(x)$$

$$b)\hspace{.1em}\frac{4}{7}\text{cos}^{-1}\left(\frac{x}{4}\right)=π$$

Hint: Consider the range of arccos, check all solutions.


#5:

Instructions: Solve each equation for exact solutions.

This identity will speed up your work. Try solving both ways.

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

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

This identity will speed up your work. Try solving both ways.

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

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


Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}\frac{π}{3}$$

$$b)\hspace{.1em}-\frac{π}{6}$$


#2:

Solutions:

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

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


#3:

Solutions:

$$a)\hspace{.1em}x=π - 3$$

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


#4:

Solutions:

$$a)\hspace{.1em}x=\frac{4}{5}$$

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


#5:

Solutions:

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

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