In this section, you will learn how to solve problems on trigonometric ratios.

Before, we look at the problems on trigonometric ratios, we have to be knowing the rule SOHCAHTOA.

SOHCAHTOA is the shortcut to remember the trigonometric ratios sin, cos and tan.

Let us see, how this shortcut works to remember the above mentioned trigonometric ratios.

Before we discuss this shortcut, let us know the name of each side of a right triangle from the figure given below.

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.

The trigonometric ratios csc θ, sec θ and cot θ are the reciprocals of sin θ, cos θ and tan θ respectively.

**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,

After having gone through the stuff given above, we hope that the students would have understood how to solve problems on trigonometric ratios.

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