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 either of the
following sets of information is given:
The length of one side and the measure of one acute
lengths of two sides
Either situation might appear on the Math IIC, so we’ll
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
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
Now you know that sin A
but you are looking to figure out the value of
, not sin A
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
your calculator has inverse trigonometric function buttons labeled
. 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
out this operation will tell you which angle between 0º and 90º
has a sine of 5/
You can solve for
by using the cos–1
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.
General Rules of Solving Right Triangles
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:
The trigonometric functions
knowledge that the sum of the angles of a triangle is 180º.
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