complementary angles are defined with respect to the addition of two angles. If the sum of two angles is 180 degrees then they are said to be supplementary angles, which forms a linear angle together. Whereas if the sum of two angles is 90 degrees, then they are said to be complementary angles, and they form a right angle together.
When two line segments or lines meet at a common point (called vertex), at the point of intersection an angle is formed. When a ray is rotated about its endpoint, then the measure of its rotation in an anti-clockwise direction is the angle formed between its initial and final position.
In fig. 1 if the ray <nobr>OP−→−</nobr> is rotated in the direction of the ray <nobr>OQ−→−</nobr>, then the measure of its rotation represents the angle formed by it. In this case, the measure of rotation that is the angle formed between the initial side and the terminal side is represented by Ɵ.
When the sum of two angles is 90°, then the angles are known as complementary angles. In other words, if two angles add up to form a right angle, then these angles are referred to as complementary angles. Here we say that the two angles complement each other.
Suppose if one angle is x then the other angle will be 90<sup>o</sup> – x. Hence, we use these complementary angles for trigonometry ratios, where on ratio complement another ratio by 90 degrees such as;
sin (90°- A) = cos A and cos (90°- A) = sin Atan (90°- A) = cot A and cot (90°- A) = tan Asec (90°- A) = cosec A and cosec (90°- A) = sec A
Hence, you can see here the trigonometric ratio of the angles gets changed if they complement each other.
The measure of angle BOD is 60<sup>o</sup> and angle AOD measures 30<sup>o</sup>. On adding both of these angles we get a right angle, therefore ∠BOD and ∠AOD are complementary angles.
The complementary to each other as the measure of the sum of both the angles is 90<sup>o</sup>. ∠POQ and ∠ABC are complementary and are called complements of each other.