# Trigonometric Ratios

In trigonometry, trigonometric ratios are derived from the sides of a right-angled triangle. There are six 6 ratios such as sine, cosine, tangent, cotangent, cosecant, and secant. You will learn here to build a trigonometry table for these ratios for some particular angles, such as 0 °, 30 °, 45 °, 60 °, 90°. There are many trigonometry formulas and trigonometric identities, which are used to solve complex equations in geometry.

## Trigonometric Ratios Definition

It is defined as the values of all the trigonometric function based on the value of the ratio of sides in a right-angled triangle. The ratios of sides of a right-angled triangle with respect to any of its acute angles are known as the trigonometric ratios of that particular angle. Consider a right-angled triangle, right-angled at B.

With respect to ∠C, the ratios of trigonometry are given as:

• sine: Sine of an angle is defined as the ratio of the side opposite(perpendicular side) to that angle to the hypotenuse.
• cosine: Cosine of an angle is defined as the ratio of the side adjacent to that angle to the hypotenuse.
• tangent: Tangent of an angle is defined as the ratio of the side opposite to that angle to the side adjacent to that angle.
• cosecant: Cosecant is a multiplicative inverse of sine.
• secant: Secant is a multiplicative inverse of cosine.
• cotangent: Cotangent is the multiplicative inverse of the tangent.

The above ratios are abbreviated as sin, cos, tan, cosec, sec and tan respectively in the order they are described. So, for Δ ABC, the ratios are defined as:

sin ∠C = (Side opposite to ∠C)/(Hypotenuse) = AB/AC

cos ∠C = (Side adjacent to ∠C)/(Hypotenuse) = BC/AC

tan ∠C = (Side opposite to ∠C)/(Side adjacent to ∠C) = AB/AC = sin ∠C/cos ∠C

cosec ∠C= 1/sin ∠C = (Hypotenuse)/ (Side Opposite to ∠C) = AC/AB

sec∠A = 1/cos ∠C = (Hypotenuse)/ (Side Opposite to ∠C) = AC/BC

cot ∠C = 1/tan ∠C = (Side adjacent to ∠C)/(Side opposite to ∠C)= BC/AB

In right Δ ABC, if ∠and ∠C are assumed as 30° and 60°, then there can be infinite right triangles with those specifications but all the ratios written above for ∠C in all of those triangles will be same. So, all the ratios for any of the acute angles (either ∠A or ∠C) will be the same for every right triangle. This means that the ratios are independent of lengths of sides of the triangle.

## Trigonometric Ratios Table

Below is the table where each ratios values are given with respect to different angles, particularly used in calculations.

 Angle 0° 30° 45° 60° 90° Sin∠C 0 1/2 1/√2 √3/2 1 Cos∠C 1 √3/2 1/√2 1/2 0 Tan∠C 0 1/√3 1 √3 ∞ Cot∠C ∞ √3 1 1/√3 0 Sec∠C 1 2/√3 √2 2 ∞ Cosec∠C ∞ 2 √2 2/√3 1

### Trigonometry Applications

Trigonometry is one of the most important branches of mathematics. Some of the applications of trigonometry are:

• Measuring the heights of towers or big mountains
• Determining the distance of the shore from the sea
• Finding the distance between two celestial bodies
• Determining the power output of solar cell panels at different inclinations
• Representing different physical quantities such as mechanical waves, electromagnetic waves, etc.