Angles are formed when two lines meet at a common point, known as a vertex in mathematics. It is represented by the symbol ∠ and is part of our day-to-day lives. The word angle comes from the Latin word "angulus,” which means a “corner.”
There are many types of angles. The complementary angles definition includes either two angles whose sum is 90 degrees. Understanding complementary angles is crucial in math and trigonometry, as you can easily calculate unknown angles within a right triangle by leveraging the relationship between sine and cosine functions of complementary angles. It can be utilised to solve real-world problems in surveying, architecture, and navigation.
Many students confuse complementary angles and supplementary angles. The difference between the two is that in complementary angles, the sum of two angles is 90 degrees, whereas in supplementary angles, the sum of two angles is 180 degrees. Essentially, complementary angles form a right angle, whereas supplementary angles form a straight line. An angle that is 30 degrees and an angle that is 60 degrees make a complementary angle together. Refer to the below image for a clear idea:
Now we know that the sum of any two complementary angles is always 90 degrees, and each of the angles “complements” the other. You can measure complementary angles by subtracting them from 90 degrees. The complement of x° is 90° – x°.
There are two types of complementary angles in geometry: adjacent complementary angles and non-adjacent complementary angles. Refer to the table below for a better understanding:
To find complementary angles of a given angle value, you need to subtract that from 90 degrees. For example, the complement of 20° is 90° - 20° = 70°. Let’s take a closer look.
Since each angle in the complementary angle pair is supposed to “complement” each other, it is easy to utilise that and find the value of the other angle if you know the value of one. Suppose the angle given is y°. To find its complement, you have to subtract its value from 90°. The formula for finding complementary angles then is:
Complement of y° = (90 - y)°
Let’s look at a few examples of complementary angles to understand how to measure complementary angles better.
Solution: To be a complementary angle, the sum of two angles needs to be 90 degrees.
46° + 44° = 90°
Thus, 46° and 44° are complementary angles.
We know that the complement of an angle measuring x° = (90 - x)°.
Here, x° = 61°
Complement of 61° = (90 - 61)° = 29°
Thus, the complement of 61° = 29°
The concept of complementary angles in trigonometry is pivotal for solving various problems and aiding in the analysis of geometric shapes and measurements. Suppose if one angle is x, then the other angle will be 90° – x. Hence, we use the trigonometric ratios of complementary angles, where one ratio is complementary to the other ratio by 90°, such as:
Complementary angles are widely used in trigonometry to solve real-world problems. Calculating distance and bearings in surveying or maritime navigation often involves the use of complementary angles in trigonometry. When designing structures like buildings or bridges, engineers use complementary angles in trigonometry to calculate angles and force acting on components. In physics, projectile motion problems often require trigonometric ratios of complementary angles to determine the trajectory of an object.
A pretty common misconception about complementary angles is that they must be adjacent or next to each other to add up to 90 degrees and be considered complementary. In reality, any two angles that sum up to 90 degrees, regardless of their position, can be considered complementary.
Another frequent error students make is confusing complementary angles and supplementary angles. In complementary angles, the value of two angles adds up to 90 degrees, whereas in supplementary angles, the value of two angles adds up to 180 degrees.
Another common misconception is that complementary angles only exist within a right angle. It is incorrect. Any two angles can be complementary angles if they add up to 90 degrees.
Complementary angle is an important concept in mathematics that can be utilised to solve complex problems. Practice questions on how to find complementary angles and further read on angle relationships and trigonometry to excel at the concept.
(Session 2025 - 26)