Math Doubts

Quotient identities

Formulas

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

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

A mathematical relation of two trigonometric functions in quotient form with another trigonometric function is called the quotient trigonometric identity.

Introduction

A trigonometric function is appeared with another trigonometric function in division form in some cases but it is not always possible to divide a trigonometric function by another trigonometric function. However, there are two possible cases in which the quotient of two trigonometric functions is also a trigonometric function.

The two possible cases are used as formulas in trigonometry. They are called the quotient trigonometric identities and simply called as quotient identities.

right triangle

When the angle of a right triangle is represented by theta. The sine, cosine, tangent and cotangent functions are written as $\sin{\theta}$, $\cos{\theta}$, $\tan{\theta}$ and $\cot{\theta}$ respectively.

Sine by Cosine Identity

The quotient of sine by cosine at an angle is equal to the tangent at that angle.

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

Cosine by Sine Identity

The quotient of cosine by sine at an angle is equal to the cotangent of that angle.

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

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