# Trigonometric Ratios of Complementary Angles

We know that complementary angles are the set of two angles such that their sum is equal to 90°. For example 30° and 60° are complementary to each other as their sum is equal to 90°.

Thus, two angles X and Y are complementary if,

∠X + ∠Y = 90°

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

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

The given triangle ∆ABC is right-angled at B; ∠A and ∠C form a complementary pair.

⇒ ∠A + ∠C = 90°

The relationship between the acute angle and the lengths of sides of a right-angle triangle is expressed by trigonometric ratios.

For the given right angle triangle, the  trigonometric ratios of ∠A is given as follows:

sin A = $\frac {BC}{AC}$

cos A = $\frac {AB}{AC}$

tan A = $\frac {BC}{AB}$

csc A = $\frac {1}{sin~A}$  = $\frac {AC}{BC}$

sec A = $\frac {1}{cos~A}$ =  $\frac {AC}{AB}$

cot A = $\frac {AB}{BC}$

The trigonometric ratio of the complement of ∠A i.e., ∠C can be given as:

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

Therefore,

sin (90°- A) = $\frac {AB}{AC}$

cos (90°- A) = $\frac {BC}{AC}$

tan (90°- A) =  $\frac {AB}{BC}$

csc (90°- A) = $\frac {1}{sin~(90°~-~ A)}$ =  $\frac {AC}{AB}$

sec (90°- A) = $\frac {1}{cos~(90°~-~ A)}$ =  $\frac {AC}{BC}$

cot (90°- A) =  $\frac {BC}{AB}$

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

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

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

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

These relations are valid for all the values of A lying between 0° and 90°. It must also be noted that sec 90°, csc 0° and tan 90° are not defined.

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

## Examples

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 if it is given that tan 2A = cot(A – 30°) and 2A is an acute angle.

Solution: Using the trigonometric ratio for complementary angles,

cot (90°- A) = tan A

⇒ tan 2A = cot (90°- 2A)

Substituting this value in the given example;

cot (90°- 2A) = cot (A – 30°)

⇒ 90°- 2A = A – 30°

⇒ A = 40°

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

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