To find angles, we can use what are known as inverse trigonometric functions.  On your calculator, the inverse trig functions will appear as \(SI{N^{ - 1}}\), \(CO{S^{ - 1}}\), or \(TA{N^{ - 1}}\). The output of these functions should always be understood as angles.  Use the following procedure to find angles:

1. Compute the sine, cosine, or tangent of the given angle (whichever is most convenient for you).
2. Evaluate the inverse trig function for that number.

More information on what sine, cosine and tangent functions are you can find in this article Finding missing sides of triangles.

Example:  Find the measure of the indicated angle to the nearest degree.

Solution:  Let’s compute the cosine of the angle.  We have

\(\large \cos \left( ? \right) = \frac{{48}}{{52}} \approx 0.923\)

Now we use the inverse cosine button on the calculator to compute

\({\cos ^{ - 1}}0.923 \approx 22.631^\circ \)

We round this result to \(23^\circ \). Now let’s try one more example:

Example:  Find the measure of the indicated angle to the nearest degree.

Solution:  This time, just to be different, let’s compute the sine of the given angle.  We have

\(\large \sin \left( ? \right) = \frac{{54}}{{90}} = 0.6\)

Since we used sine, we will compute the inverse sine of 0.6. We have

\({\sin ^{ - 1}}0.6 \approx 36.870\)

We round this result to \(37^\circ \).

Below you can download some free math worksheets and practice.