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.

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