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 ?

Here is the answer.

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**

**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

adjacent side = 12

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,

AB^{2} = BC^{2} + CA^{2}

AB^{2} = 7^{2} + 24^{2}

AB^{2} = 49 + 576

AB^{2} = 625

AB^{2} = 25^{2}

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,

AC^{2} = AB^{2} + BC^{2}

15^{2} = AB^{2} + 12^{2}

225 = AB^{2} - 144

225 - 144 = AB^{2}

81 = AB^{2}

9^{2 }= AB^{2}

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,

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