**Basic Trigonometric Ratios : **

We know that six ratios can be formed using the three lengths a, b, c of sides of a right triangle ABC.

Interestingly, these ratios lead to the definitions of six basic trigonometric functions.

First, let us recall the trigonometric ratios which are defined with reference to a right triangle.

sin θ = opposite side / hypotenuse

cos θ = adjacent side / hypotenuse

With the help of sin θ and cos θ, the remaining trigonometric ratios tan θ, cot θ, csc θ and sec θ are determined by using the relations

tan θ = sin θ / cos θ

csc θ = 1 / sin θ

sec θ = 1 / cos θ

cot θ = cos θ / sin θ

And also,

sin θ = 1 / csc θ

cos θ = 1 / sec θ

**Note : **

1. sin θ and csc θ are reciprocal to each other.

2. cos θ and sec θ are reciprocal to each other.

3. tan θ and cot θ are reciprocal to each other.

**Example : **

1. If sin θ = 3/5, then csc θ = 5/3.

2. If cos θ = 4/5, then sec θ = 5/4.

3. If tan θ = 3/4, then cot θ = 4/3.

**Problem 1 :**

In the right triangle PQR given below, find the basic trigonometric ratios of the angle θ.

**Solution :**

In the triangle shown above, for the angle θ,

opposite side = 5

adjacent side = 12

hypotenuse = 13

Then, the basic trigonometric ratios of the angle θ are

sin θ = 5/13 cos θ = 12/13 tan θ = 5/12 |
csc θ = 13/5 sec θ = 13/12 cot θ = 5/12 |

**Problem 2 :**

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

In the triangle shown above, 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

In the triangle shown above, for the angle θ,

opposite side = 7

adjacent side = 24

hypotenuse = 25

Then, the basic trigonometric ratios of the angle θ are

sin θ = 7/25 cos θ = 24/25 tan θ = 7/24 |
csc θ = 25/7 sec θ = 25/24 cot θ = 24/7 |

**Problem 3 :**

If sin θ = 13/85 and cos θ = 84/85, then find the values of tan θ and cos θ.

**Solution : **

Finding the value of tan θ :

tan θ = sin θ / cos θ

tan θ = (13/85) / (84/85)

tan θ = (13/85) ⋅ (85/84)

tan θ = (13 ⋅ 85) / (85 ⋅ 84)

tan θ = 13/84

Finding the value of cot θ :

cot θ = 84/13

After having gone through the stuff given above, we hope that the students would have understood how to find the values of basic trigonometric ratios using right triangle.

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