# RECIPROCAL RELATION OF TRIGONOMETRIC RATIOS

Reciprocal relation of trigonometric ratios :

In the six trigonometric ratios sin, cos, tan, csc, sec and cot, there is a reciprocal relation among them.

Here, the pairs of trigonometric relations are given between which we have reciprocal relation.

sin θ <---------> csc θ

cos θ <---------> sec θ

tan θ <---------> cot θ

More clearly, To have better understanding on reciprocal relations of trigonometric ratios, first we have to know the shortcut SOHCAHTOA which is related to the trigonometric ratios sin, cos and tan

To understand the shortcut, first we have to divide SOHCAHTOA in to three parts as given below. What do SOH, CAH and TOA stand for ?   From the above figures, we can derive formulas for the three trigonometric ratios sin, cos and tan as given below. csc θ  =  Hypotenuse / Opposite side sec θ = Hypotenuse / Adjacent side cot θ = Adjacent side/Opposite side

## Reciprocal relation of trigonometric ratios - Practice problems

Problem 1 :

In the right triangle shown below, find the six trigonometric ratios of the angle θ. Solution :

From the triangle shown above,

opposite side  =  5

hypotenuse  =  13

Therefore, Problem 2 :

In the right triangle shown below, find the six trigonometric ratios of the angle θ. Solution :

From the right triangle shown above,

AC  =  24

BC  =  7

By Pythagorean Theorem,

AB2  =  BC2 + CA2

AB2  =  72 + 242

AB2  =  49 + 576

AB2  =  625

AB2  =  252

AB  =  25

Now, we can use the three sides to find the six trigonometric ratios of angle θ.

Therefore, Problem 3 :

In triangle ABC, right angled at B, 15 sin A = 12. Find the other five trigonometric ratios of the angle A. Also find the six ratios of the angle C.

Solution :

Given :  15sin A  =  12.

Then,

sin A  =  12/15

Therefore,

opposite side  =  12

hypotenuse  =  15

Let us consider the right triangle ABC where right angled at B, with

BC  =  12

AC  =  15 By Pythagorean theorem,

AC2  =  AB2 + BC2

152  =  AB2 + 122

225  =  AB2 - 144

225 - 144  =  AB2

81  =  AB2

92  =  AB2

9  =  AB

Now, we can use the three sides to find the five trigonometric ratios of angle A and six trigonometric ratios of angle C.

Therefore,  After having gone through the stuff given above, we hope that the students would have understood the reciprocal relation of trigonometric ratios.

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