Sine by Cosine Quotient identity

Formula

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

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

Introduction

In the trigonometric mathematics, the sine, cosine and tangent functions are defined mathematically at an angle in the ratio form of the sides of a right triangle. Actually, it is possible to divide the sine by cosine and their quotient is equal to the tangent. Hence, the mathematical relation between them is called the sine by cosine quotient identity.

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

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

Mathematically, the quotient of sine by cosine is equal to tangent and it is expressed in the following mathematical form.

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

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

Usage

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

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

Other forms

The angle in sine by cosine quotient trigonometric identity can be denoted by any symbol. Hence, the sine by cosine quotient identity is written in three ways popularly.

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

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

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

Proof

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