# Reciprocal identity of Sin function

## Formula

$\csc{\theta} \,=\, \dfrac{1}{\sin{\theta}}$

### Proof

Sine is a ratio of lengths of opposite side to hypotenuse and the cosecant is a ratio of lengths of hypotenuse to opposite side. The sine and cosecant functions are mutually reciprocals. Therefore, the reciprocal of sin of angle is equals to cosecant of angle.

$\Delta RPQ$ is a right triangle and its angle is theta.

#### Write sine in its ratio form

Firstly, write sin of angle theta ($\sin{\theta}$) in its ratio form.

$\sin{\theta} \,=\, \dfrac{QR}{PQ}$

#### Write cosecant in its ratio form

Similarly, write the cosecant of angle theta ($\csc{\theta}$ or $\operatorname{cosec}{\theta}$) in its ratio form.

$\csc{\theta} \,=\, \dfrac{PQ}{QR}$

#### Relation between sine and cosecant

The value of cosecant function in its ratio form can be written in reciprocal form to prove the relation between sine and cosecant functions mathematically.

$\implies \csc{\theta} \,=\, \dfrac{1}{\dfrac{QR}{PQ}}$

$\,\,\, \therefore \,\,\,\,\,\,$ $\csc{\theta} \,=\, \dfrac{1}{\sin{\theta}}$

Therefore, it is proved that the reciprocal of sin function is equal to cosecant function. It will be used as a formula in trigonometric mathematics.

###### Note

The angle of right triangle can be represented by any symbol and the reciprocal identity of sin function should be written in the respective angle.

For example, if $x$ is angle of right triangle, then

$\csc{x} \,=\, \dfrac{1}{\sin{x}}$

Similarly, if $A$ is angle of right triangle, then

$\csc{A} \,=\, \dfrac{1}{\sin{A}}$

The reciprocal identity of sin function is written in this form but the only changing factor is angle of right triangle.

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