Algebra
Textbooks
Boundless Algebra
Trigonometry
Trigonometry and Right Triangles
Algebra Textbooks Boundless Algebra Trigonometry Trigonometry and Right Triangles
Algebra Textbooks Boundless Algebra Trigonometry
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 4
Created by Boundless

Finding Angles From Ratios: Inverse Trigonometric Functions

The inverse trigonometric functions can be used to find the acute angle measurement of a right triangle.

Learning Objective

  • Recognize the role of inverse trigonometric functions in solving problems about right triangles


Key Points

    • A missing acute angle value of a right triangle can be found when given two side lengths. 
    • To find a missing angle value, use the trigonometric functions sine, cosine, or tangent, and the inverse key on a calculator to apply the inverse function ($\arcsin{}$, $\arccos{}$, $\arctan{}$), $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$.

Full Text

Using the trigonometric functions to solve for a missing side when given an acute angle is as simple as identifying the sides in relation to the acute angle, choosing the correct function, setting up the equation and solving.  Finding the missing acute angle when given two sides of a right triangle is just as simple.

Inverse Trigonometric Functions

In order to solve for the missing acute angle, use the same three trigonometric functions, but, use the inverse key ($^{-1}$on the calculator) to solve for the angle ($A$) when given two sides.

$\displaystyle{ A^{\circ} = \sin^{-1}{ \left( \frac {opposite}{hypotenuse} \right) } }$

$\displaystyle{ A^{\circ} = \cos^{-1}{ \left( \frac {adjacent}{hypotenuse} \right) } }$

$\displaystyle{ A^{\circ} = \tan^{-1}{\left(\frac {opposite}{adjacent} \right) }}$

Example 1:  For a right triangle with hypotenuse length $25~\mathrm{feet}$, and opposite side length $12~\mathrm{feet}$, find the acute angle to the nearest degree:

Right triangle

Find the measure of angle $A$, when given the opposite side and hypotenuse.

From angle $A$, the sides opposite and hypotenuse are given.  Therefore, use the sine trigonometric function. (Soh from SohCahToa)  Write the equation and solve using the inverse key for sine.

$\displaystyle{ \begin{aligned} \sin{A^{\circ}} &= \frac {opposite}{hypotenuse} \\ \sin{A^{\circ}} &= \frac{12}{25} \\ A^{\circ} &= \sin^{-1}{\left( \frac{12}{25} \right)} \\ A^{\circ} &= \sin^{-1}{\left( 0.48 \right)} \\ A &=29^{\circ} \end{aligned} }$

[ edit ]
Edit this content
Prev Concept
Sine, Cosine, and Tangent
Radians
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.