Math Doubts

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 at an angle of a right triangle and their quotient is always equal to the tangent. Hence, this mathematical relation is called the sine by cosine quotient identity.

right triangle

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.

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

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.

Other forms

The quotient of sine by cosine trigonometric identity can be expressed in terms of any variable angle.

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

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

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

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.

Proof

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

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