This is a way of remembering how to compute the values the trigonometric ratios sin, cosine and tangent of an angle.

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 PQR given below, find the six trigonometric ratios of the angle θ.

**Solution :**

From the figure given above,

opposite side = 5

adjacent side = 12

hypotenuse = 13

Therefore,

**Problem 2 :**

From the figure given below, find the six trigonometric ratios of the angle θ.

**Solution : **

From the figure given above, AC = 24 and 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 that 15 sin A = 12, so sin A = 12 / 15

Therefore, opposite side = 12 and hypotenuse = 15

Let us consider the triangle ABC where right angled at B, with BC = 12 and AC = 15.

By Pythagorean theorem,

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

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

225 = AB^{2} + 144

Subtract 144 from each side.

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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