Inverse Trigonometric Functions

Complementary angles form an important part of geometry. You may already know that the sum of their measurement determines the complement and supplement of two angles. 

Two angles with a sum of 90 degrees are called complementary angles. In other words, if two angles have a sum equivalent to a right angle’s measurement, they are complementary angles.

For example, consider the two angles, 50°, and 40°. Since they add up to 90°, they are complementary angles. 

Definition of Complementary Angles

Two angles are complementary if their sum adds up to 90°, i.e., they make a right angle when they are put together. Each angle of the complementary pair is called the complement of the other angle. 

If Angle x + Angle y = 90°, then Angle x and Angle y are complements of each other, i.e., complementary angles.

Finding a Complementary Angle

Finding a complementary angle will be an easy task if you have understood the definition and meaning of complementary angles.  The most straightforward method to find a complementary angle for any angle is by subtracting them from 90 degrees.

Let us take some examples to grab a firm hold of the concept.

Example 1: Find the complementary angle of 48°.

Solution: 

As stated in the above discussion, subtract the angle given in the question from 90°.

90° – 48° = 42°

Hence, the complementary angle of 48° is 42°.

Example 2: What is the complementary angle of 89°?

Solution:

90° – 89° = 1°

Therefore, the complementary angle of 89° is 90°.

Types of Complementary Angles

There are two types of complementary angles: Adjacent complementary angles and non-adjacent complementary angles.

Adjacent Complementary Angles

When two complementary angles have a common arm and vertex, they are adjacent complementary angles. In other words, if two angles have a sum of 90° with a common arm and vertex, they are known as adjacent complementary angles.

Non-Adjacent Complementary Angles

Defining them is even easier. When two complementary angles are not adjacent complementary angles, they are known as non-adjacent complementary angles. 

Non-adjacent complementary angles do not have a common vertex or arm. However, since they are complementary angles, their sum adds up to 90°, forming a right angle.

Properties of Complementary Angles

Having read the definition of complementary angles and their types, now it is time to learn about their properties. The following are the properties of complementary angles:

• Two angles can be complementary only if their sum adds up to 90°

• Complementary angles may be adjacent or non-adjacent

• In the case of two complementary angles, one angle complements the other

• Only two angles can be complementary

• Even if their sum is 90°, three or more angles cannot be complementary

• In the case of a right-angled triangle, two acute angles are complementary

• Though a pair of complementary angles is always acute, all pairs of acute angles are not complementary

Trigonometric Ratios of Complementary Angles

We know that two angles are complementary if their sum is 90°. It means that for an acute angle θ, θ and (90°−θ) are complementary angles. In right-angled triangles, the sine and cosine of angles are the ratios of sides.

It is worth noting that:

• In a right-angled triangle, an angle’s sine corresponds to the ratio of the side opposite to the hypotenuse

• In a right-angled triangle, an angle’s cosine corresponds to the ratio of the side that is adjacent to the hypotenuse

Consider the above right-angled triangle, PQR. As you already know, three angles in a triangle sum up to a total of 180°. Therefore, m∠P + m∠Q + m∠R = 180°. Since ∠Q is a right angle, m∠Q=90°, which means m∠P + m∠R = 90°.

Now, because the acute angles total 90°, it is clear that they are complementary.

Further, for the reference angle θ:

sin θ = QRPR

cos θ = PQPR

tan θ = QRPQ

cosec θ = PRQR

sec θ = PRPQ

cot θ = PQQR

And for the reference angle 90°−θ:

sin (90°−θ) = PQPR 

cos (90°−θ) = QRPR

tan (90°−θ) = PQQR

csc (90°−θ) = PRPQ

sec(90°−θ) = PRQR

cot(90°−θ) = QRPQ

Now, we can easily obtain:

sin (90°−θ) = cos θ

tan (90°−θ) = cot θ

sec (90°−θ) = cosec θ

cos (90°−θ) = sin θ

cot (90°−θ) = tan θ

cosec (90°−θ) = sec θ

It is important to note:

• In the case of an acute angle, sine is equal to the cosine of its complement and vice versa

• Any acute angle’s cotangent is equal to the tangent of its complement and vice versa

• Similarly, the cosecant of an acute angle is equivalent to the secant of its complement and vice versa

Conclusion

Two angles can be complementary angles if they sum up to 90 degrees. Any change from that number would cease them from being complementary angles. In essence, it means that the sum of complementary angles totals that of a right angle. 

When there are two complementary angles, both are complements of each other.  It is worth noting that three or more angles can never be complementary, even if their sum is 90 degrees. 

It is advisable to read about ‘Angles’ holistically to enhance your knowledge about the topic.