Complementary angles

If a fifth grader asks you, what are complementary angles, you need not stare at him perplexed. This short note would give you enough information to answer that question.
Complementary angles definition is not as complicated as the name sounds. Take a look at the following figure:

We have two angles here. One is 40 degrees and the other is 50 degrees. But that is not important. The important point to be noted here is that 40 + 50 = 90 degrees. That means that the measure of these two angles add up to 90 degrees. With this understanding let us now write the definition of complementary-angles.

Definition: Complementary angles:

Two angles, whose sum is 90 degrees are said to be complementary-angles. The angles may or may not be adjacent angles. For example, see the figure below:

In this figure the two angle measures are 27 degrees and 63 degrees. Sum of these measures is 27 + 63 = 90 degrees. Therefore we can say that these two are a pair of complementary-angles. Examples of other such pair could be: 30 – 60, 45 – 45, 15 – 75, etc. There can be infinite such pairs. When a pair of angles are complementary-angles, we say that one is the compliment of the other, or one compliments the other. A pair of complimentary angles is always acute. That is to say that both the angles have to be acute angles.

Where do complementary angles occur?

Complementary angles are most common in a right triangle. In a right triangle since one angle is 90 degrees, the sum of the other two angles has to be 90 degrees. That is because we know that sum of all the three angles in a triangle should always be 180 degrees. Therefore in a right triangle, the two smaller angles are compliments of each other, or in other words they are complementary-angles.

The concept of complementary angles also finds its application in trigonometry. The sine ratio of an angle is always equal to the cosine ratio of the compliment of that angle. Similarly the tangent ratio of an angle is always equal to the cotangent ratio of the compliment of that angle. Also the secant ratio of an angle would be equal to the cosecant ratio of the compliment of that angle.