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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