Lesson: Chapter - 9

Solving Right Triangles

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 you are given either of the following sets of information:

1. The length of one side and the measure of one acute angle
2. The lengths of two sides

Given: One Angle and One Side

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: < B = 180 - 90 - 35 = 55^o

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.

Given: Two Sides

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

Now you know that sin A = ⁵/₈, but you are trying to find out the value of <A , not sin A. To do this, you need to use some standard algebra and isolate <A . 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 exactly which angle between 0º and 90º has a sine of ⁵/₈.

You can solve for <B 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.

General Rules of Solving Right Triangles

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

1. The trigonometric functions
2. The Pythagorean theorem
3. The knowledge that the sum of the angles of a triangle is 180º

There is no “right” way to solve a right triangle. One way that is usually wrong, however, is solving for an angle or a side in the first step, approximating that measurement, and then using that approximation to finish solving the triangle. This approximation will lead to inaccurate answers, which in some cases might mean that your answer will not match the answer choices.

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Mathematics Practice Questions