In Mathematics, the complementary angles are theÂ set of two angles such thatÂ their sum is equal to 90Â°. For example, 30Â° and 60Â° are complementary to each other as their sum is equal to 90Â°. In this article, let us discuss in detail about the complementary angles and the trigonometric ratios of complementary angles with examples in a detailed way.

## Complementary Angles Definition

The two angles, say âˆ X and âˆ Y are complementary if,

âˆ X + âˆ Y = 90Â°

In such a condition âˆ X is known as the complement of âˆ Y and vice-versa.

In a right angle triangle, as the measure of the right angle is fixed, the remaining two angles always form the complementary as the sum of angles in a triangle is equal to 180Â°.

## Finding Trigonometric Ratios of Complementary Angles

Assume a triangle âˆ†ABC, which is right-angled at B.

âˆ A and âˆ C form a complementary pair.

â‡’ âˆ A + âˆ C = 90Â°

The relationship between the acute angle and the lengths of sides of a right-angle triangle is expressed by trigonometric ratios.Â For the given right angle triangle, the Â trigonometric ratios of âˆ A is given as follows:

sin A = BC/AC

cos A = AB/AC

tan A =BC/AB

csc A = 1/sin A = AC/BC

sec A =1/cos A = AC/AB

cot A = 1/tan A = AB/BC

The trigonometric ratio of the complement of âˆ A. It means that the âˆ C can be given as 90Â° – âˆ A

As âˆ C = 90Â°- A (A is used for convenience instead of âˆ A ), and the side opposite to 90Â° – A is AB and the side adjacent to the angle 90Â°- A is BC as shown in the figure given above.

Therefore,

sin (90Â°- A) = AB/AC

cos (90Â°- A) = BC/AC

tan (90Â°- A) = AB/BC

csc (90Â°- A) =1/sin (90Â°- A) = AC/AB

sec (90Â°- A) = 1/cos (90Â°- A) = AC/BC

cot (90Â°- A) = 1/tan (90Â°- A)Â = BC/AB

Comparing the above set of ratios with the ratios mentioned earlier, it can be seen that;

sin (90Â°- A) = cos A ; cosÂ (90Â°- A) = sin A

tan (90Â°- A) = cot A; cotÂ (90Â°- A) = tan A

sec (90Â°- A) = csc A; cscÂ (90Â°- A) = sec A

These relations are valid for all the values of A that lies between 0Â°Â and 90Â°.

**Summary:**

- Sin of an angle = Cos of its complementary angle
- Cos of an angle = Sin of its complementary angle
- Tan of an angle = Cot of its complementary angle

### Trigonometric Ratios of Complementary Angles Examples

To have a better insight on trigonometric ratios of complementary angles consider the following example.

**Example: **

If A, B and C are the interior angles of a right-angle triangle, right-angled at B then find the value of A, given that tan 2A = cot(A – 30Â°) and 2A is an acute angle.

**Solution: **

Using the trigonometric ratio of complementary angles,

cotÂ (90Â°- A) = tan A

From this ratio, we can write the above expression as:

â‡’ tan 2A = cotÂ (90Â°- 2A) ….(1)

Given expression is tan 2A = cot (A – 30Â°) …(2)

Now, equate the equation (1) and (2), we get

cot (90Â°- 2A) = cot (A – 30Â°)

â‡’Â 90Â°- 2A = A –Â 30Â°

â‡’3A =Â 90Â° + 30Â°

â‡’3A =Â 120Â°

â‡’A =Â 120Â°/ 3

â‡’ A = 40Â°

Thus, the measure of the acute angle A can be easily calculated by making use of trigonometry ratio of complementary angles.

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