
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:
 The length of one side and the measure of one acute angle
 The lengths of two sides
Either situation might appear on the Math IC, so we cover
both.
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:
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.
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 trying to find out the value of , not sin A. To
do this, you need to use some standard algebra and isolate . In other words, you have to find
the inverse sine of both sides of the equation sin A = ^{5}⁄_{8}.
Luckily, your calculator has inversetrigonometricfunction 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 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 inversetrigonometricfunction
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:
 The trigonometric functions
 The Pythagorean theorem
 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.
