Quotient identity of Trigonometric functions cosine and sine

Formula

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

Proof

right angled triangle

$\Delta BAC$ is a right angled triangle and the angle of the triangle is theta ($\theta$). Now, write the sine and cosine functions in mathematical form in terms of sides of the triangle.

$\sin \theta$ $\,=\,$ $\dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$

$\cos \theta$ $\,=\,$ $\dfrac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}$

$BC$, $AB$ and $AC$ are the length of opposite side, adjacent side and hypotenuse respectively in $\Delta CAB$. Now, express sine and cosine functions in terms of lengths of the sides of the right angled triangle.

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

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

Now, divide the cosine function by sine function to study the quotient property of cosine and sine functions in trigonometry.

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

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

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

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

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

$AB$ and $BC$ represent the length of adjacent side and opposite side in this right angled triangle.

$\implies \dfrac{\cos \theta}{\sin \theta}$ $\,=\,$ $\dfrac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side}$

The ratio of length of adjacent side to opposite side is cotangent of the angle of the triangle and it is expressed as $\cot \theta$ in mathematics.

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

The quotient of cosine of angle by the sine of angle is equal to the cotangent of the angle. Hence, the trigonometric law is called the quotient rule of cosine and sine. It is mainly used to write the ratio of cosine of angle to sine of angle as cotangent and vice-versa.

Verification

Take, the angle of right angled triangle is $30^\circ$. Therefore, the value of sine, cosine and cotangent functions for the angle $30^\circ$ are $\sin 30^\circ \,=\, \dfrac{1}{2}$, $\cos 30^\circ \,=\, \dfrac{\sqrt{3}}{2}$ and $\cot 30^\circ \,=\, \sqrt{3}$

Evaluate the ratio of cosine of angle $30^\circ$ to sine of angle $30^\circ$.

$\dfrac{\cos 30^\circ}{\sin 30^\circ} \,=\, \dfrac{\dfrac{\sqrt{3}}{2}}{\dfrac{1}{2}}$

$\implies \dfrac{\cos 30^\circ}{\sin 30^\circ} \,=\, \dfrac{\sqrt{3}}{2} \times \dfrac{2}{1}$

$\implies \dfrac{\cos 30^\circ}{\sin 30^\circ} \,=\, \dfrac{\sqrt{3} \times 2}{2 \times 1}$

$\implies \require{cancel} \dfrac{\cos 30^\circ}{\sin 30^\circ} \,=\, \dfrac{\sqrt{3} \times \cancel{2}}{\cancel{2}}$

$\implies \dfrac{\cos 30^\circ}{\sin 30^\circ} \,=\, \sqrt{3}$

The value of $\cot 30^\circ$ is equal to $\sqrt{3}$.

$\therefore \,\,\,\,\,\, \dfrac{\cos 30^\circ}{\sin 30^\circ} \,=\, \cot 30^\circ$

It represents the quotient relation of cosine and sine functions with cotangent function.

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