Math Doubts

Cosine by Sine Quotient identity

Formula

$\dfrac{\cos{\theta}}{\sin{\theta}} \,=\, \cot{\theta}$

The quotient of cosine by sine equals to cotangent is called the cosine by sine quotient trigonometric identity.

Introduction

The sine, cosine and cotangent functions are defined in trigonometric mathematics at an angle in the ratio form of the sides of a right triangle. It is possible to divide the cosine by sine mathematically and their quotient is equal to the cotangent. Hence, it is called the cosine by sine quotient identity.

right triangle

If the angle of a right triangle is represented by a symbol theta, then the sine, cosine and cot functions are written as $\sin{\theta}$, $\cos{\theta}$ and $\cot{\theta}$ respectively in mathematics. The quotient of cosine by sine is written in division form as follows.

$\dfrac{\cos{\theta}}{\sin{\theta}}$

Mathematically, the quotient of cosine by sine is equal to cotangent and it is expressed in mathematical form as follows.

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}} \,=\, \cot{\theta}$

This mathematical relation between sine, cosine and cot functions is called the cosine by sine quotient identity.

Usage

The cosine by sine quotient identity is used as a formula in two different cases in mathematics.

  1. To simplify the quotient of cosine by sine as cotangent.
  2. To express the cotangent as the quotient of cosine by sine.

Other forms

The angle in the cosine by sine quotient identity can be denoted by any symbol. Therefore, it is written in three popular mathematical forms.

$(1).\,\,\,\,\,\,$ $\dfrac{\cos{A}}{\sin{A}} \,=\, \cot{A}$

$(2).\,\,\,\,\,\,$ $\dfrac{\cos{x}}{\sin{x}} \,=\, \cot{x}$

$(3).\,\,\,\,\,\,$ $\dfrac{\cos{\alpha}}{\sin{\alpha}} \,=\, \cot{\alpha}$

Proof

Learn how to derive the cosine by sine quotient identity in mathematical form geometrically.

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