In this module, we will learn “Trigonometric Ratios of Complementary Angles”. First, we shall know what are complementary angles. Two angles are said to be complementary if their sum equals 90°.

The complementary angles are a pair of angles whose sum equals 90 degrees. The angles 30° and 60°, for example, are complimentary because their sum equals 90°.

Definition of Complementary Angles

If A + B = 90°, the two angles, say A and B, are complimentary.
A is known as the complement of B in this situation, and vice versa.

Because the measure of the right angle is fixed in a right-angle triangle, the remaining two angles always form the complementary because the sum of angles in a triangle is 180°.

Finding Trigonometric Ratios of Complementary Angles

What is a trigonometric ratio?

Trigonometric ratios express the relationship between the acute angle and the lengths of the sides of a right-angle triangle.

In triangle ABC, right-angled at B, do you see any pair of complementary angles?

Since ?A + ?C = 90°, they form such a pair.

Now let us write the trigonometric ratios for ?C = 90° – ?A.
For convenience, we shall write 90° – A instead of 90° – ?A.
What would be the side opposite and the side adjacent to the angle 90° – A?
Here AB is the side opposite and BC is the side adjacent to the angle 90° – A.
Therefore,

Now, compare the ratios of angle A and angle (90° – A).

Observe that:

Trigonometric Ratios of Complementary Angles - Examples

Example: Evaluate: sin 65° – cos 25°.

Solution: We know, sin A = cos (90° – A)

So, sin 65° = cos (90° – 65°) = cos 25°

Therefore,

Sin 65° – cos 25° = cos 25° – cos 25° = 0

Example: Express cot 75° + sin 75° in terms of trigonometric ratios of angles between 0° and 45°.

Solution:

cot 75° + sin 75° = cot (90° – 15°) + sin (90° – 15°)
                            = tan 15° + sin 15°

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