High School Geometry Unlocked (2016)
Chapter 4. Trigonometry
Lesson 4.2. Complementary Angles
In geometry, two angles are supplementary to each other if their sum equals 180°. For example, 45° is supplementary to 135°.
When two angles have a sum of 90°, they are known as complementary angles. For example, 30° is complementary to 60°. In a right triangle, the two acute angles are always complementary.
For each of the angle measures in the table below, find the measure of the complementary angle.
Check your answers at the end of the chapter.
The concept of complementary angles is very important in trigonometry. You may have noticed in the examples of the right triangles in the previous section, that the complementary angles have the same sin and cos values—except that they’re switched. That is, the sine of one angle is equal to the cosine of its complement. For example, the sine of 30° is equal to the cosine of 60°. (Both are equal to 0.5.)
Additionally, the tangents of complementary angles are reciprocals of each other. In other words, the tangent of one angle is equal to the reciprocal of the tangent of its complement. For example, the tangent of 60° is 
, while the tangent of 30° is 
.
Complementary Angles
Complementary angles have a sum of 90°.
The sine of x° is equal to the cosine of 90 − x°.
The cosine of x° is equal to the sine of 90 − x°.
The tangent of x° is the reciprocal of the tangent of 90 − x°.
In the table below, use the given value on the left to determine the indicated value on the right.
Check your answers at the end of the chapter.
Your Calculator
Scientific and graphing calculators have sin, cos, and tan built in! When given an angle, the calculator can tell you the value of sin, cos, or tan for that angle.
Want to try it? With your calculator in degree mode, enter sin(30°). The calculator should show 
, or 0.5.
You can try the different functions for 30° and 60°, and compare the results with the values in the table shown previously.
Your calculator can also do the inverse of these functions, which takes the ratio and solves for the angle. For example, the inverse of sin30° is just 30°. On your calculator, the inverse of sin might look like sin−1, or “arcsin.”
If you enter sin−1(0.5), in degree mode, your calculator should return 30°.
It is not typically
practical to solve
inverse functions
without a calculator.
The inverse function
of some “special”
right triangles (such
as 30°-60°-90° or
45°-45°-90°) can
be memorized. One
other way to solve
inverse functions is to
observe a graph of sin,
cos, or tan values.
Which function should
you use to solve?
If you have all three
side lengths of the
triangle, then all
three functions
work equally well!
What is the value of x° and y°? Use sin−1, cos−1, or tan−1.
Check your answers at the end of the chapter.