One of the most important applications of trigonometric functions is to “solve” a right triangle. By now, you should know that every right triangle has five unknowns: the lengths of its three sides and the measures of its two acute angles. Solving the triangle means finding the values of these unknowns. You can use trigonometric functions to solve a right triangle if either of the following sets of information is given:

Either situation might appear on the Math IIC, so we’ll cover both.

The right triangle below has an acute angle of 35º and a side of length 7.

To find the measure of the other acute angle, just subtract the measures of the other two angles from 180º:

To find the lengths of the other two sides, use trigonometric functions relating the given angle measure to the given side length. The key to problems of this type is to choose the correct trigonometric functions. In this question, you are given the measure of one angle and the length of the side opposite that angle, and two trigonometric functions relate these quantities. Since you know the length of the opposite side, the sine (opposite ⁄ hypotenuse) will allow you to solve for the length of the hypotenuse. Likewise, the tangent (opposite ⁄ adjacent) will let you solve for the length of the adjacent side.

You’ll need your calculator to find sin 35º and tan 35º. But the basic algebra of solving right triangles is easy.

The right triangle below has a leg of length 5 and a hypotenuse of length 8.

First, use the Pythagorean theorem to find the length of the third side:

Next, use trigonometric functions to solve for the acute angles:

Now you know that sin A = ⁵/₈, but you are looking to figure out the value of , not sin A. To do this, you need to use some standard algebra and isolate the angle. In other words, you have to find the inverse sine of both sides of the equation sin A = ⁵ /₈. Luckily, your calculator has inverse trigonometric function buttons labeled sin^–1, cos^–1, and tan^–1. These inverse trigonometric functions are also referred to as arcsine, arccosine, and arctangent.

For this problem, use the sin^–1 button to calculate the inverse sine of ⁵ /₈. Carrying out this operation will tell you which angle between 0º and 90º has a sine of ⁵/ ₈.

You can solve for by using the cos^–1 button and following the same steps. Try it out. You should come up with a value of 51.3º.

To solve this type of problem you must know the proper math, and you also have to know how to use the inverse trigonometric function buttons on your calculator. This type of question puts the “C” in Math IIC.

We’ve just shown you two of the different paths you can take when solving a right triangle. The actual solution will depend on the specific problem, but the same three tools are always used:

There is not necessarily a “right” way to solve a right triangle. One way that is usually “wrong,” however, is solving for an angle or side in the first step, approximating that measurement, and then using that approximation to finish solving the triangle. This approximation will lead to less accurate answers, which in some cases might cause your answer not to match with the answer choices.