According to the quotient of cosine by sine identity, the quotient of cosine by sine is equal to the cotangent.

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

The trigonometric quotient identity is derived geometrically in mathematical form from a right triangle. Now, it’s your time to learn how to prove the quotient of cosine by sine identity from a geometric shape right triangle.

$\Delta BAC$ is a right angled triangle. In this triangle, $\overline{BC}$, $\overline{AB}$ and $\overline{AC}$ are opposite side, adjacent side and hypotenuse respectively, and the angle of this triangle is taken as theta.

According to the definitions of the trigonometric functions, express the sine and cosine functions in ratio form at an angle theta.

$\sin{\theta} \,=\, \dfrac{BC}{AC}$

$\cos{\theta} \,=\, \dfrac{AB}{AC}$

Now, divide the cosine function by the sine function for evaluating the quotient of them.

$\dfrac{\cos{\theta}}{\sin{\theta}} \,=\, \dfrac{\dfrac{AB}{AC}}{\dfrac{BC}{AC}}$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{AC} \times \dfrac{AC}{BC}$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC} \times \dfrac{AC}{AC}$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC} \times \require{cancel} \dfrac{\cancel{AC}}{\cancel{AC}}$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC} \times 1$

$\implies$ $\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC}$

We have derived in the above step that the ratio of cosine by sine is equal to the quotient of lengths of adjacent side ($AB$) by opposite side ($BC$).

$\dfrac{\cos{\theta}}{\sin{\theta}}$ $\,=\,$ $\dfrac{AB}{BC}$

According to the trigonometry, the quotient of $AB$ by $BC$ represents the cotangent. Here, the angle of right triangle is theta. Therefore, the cot function is written as $\cot{\theta}$.

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

Therefore, it is proved that the quotient of cosine function by sine function is equal to the cotangent function mathematically.

Latest Math Topics

Apr 18, 2022

Apr 14, 2022

Apr 05, 2022

Mar 18, 2022

Mar 05, 2022

Latest Math Problems

Apr 06, 2022

Mar 22, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2021 Math Doubts, All Rights Reserved