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

Aug 31, 2024

Aug 07, 2024

Jul 24, 2024

Dec 13, 2023

Latest Math Problems

Sep 04, 2024

Jan 30, 2024

Oct 15, 2023

Copyright © 2012 - 2023 Math Doubts, All Rights Reserved