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.