Reciprocal identity of Sine and Cosecant

The reciprocal relation of sine function with cosecant function and cosecant function with sine function is called the reciprocal identity of sine and cosecant functions.

Formula

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

$(2)\,\,\,\,\,\,$ $\csc{\theta} \,=\, \dfrac{1}{\sin{\theta}}$

Introduction

Sine and Co-secant functions have reciprocal relation mutually. So, sine function can be written in terms of cosecant function in reciprocal form. Similarly, cosecant function can also be written in terms of sine function in reciprocal form. The reciprocal relationship between sine and cosecant functions is used as a formula in trigonometry.

Proof

Consider a right angled triangle ($\Delta BAC$) and its angle is theta ($\theta$) to prove the reciprocal relation for sine and cosecant functions.

right angled triangle
Defining Sine function

Write the sine function in terms of ratio of lengths of the sides of right angled triangle.

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

Writing it in Reciprocal form

Express the ratio of sides for the sine function in reciprocal form.

$\sin{\theta} \,=\, \dfrac{1}{\dfrac{Length \, of \, Hypotenuse}{Length \, of \, Opposite \, side}}$

Expressing it in terms of Co-secant

In fact, the ratio of length of the hypotenuse to length of the opposite side is called cosecant function according to the trigonometry.

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

Therefore, sine function can be expressed in terms of cosecant function on the basis of this rule.

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

The trigonometric identity represents how sine function is having relation with cosecant function in reciprocal form.

Similarly, the reciprocal relation of cosecant function with sine function can also be derived in mathematics. Repeat the same procedure once to prove it in trigonometry.

Defining Cosecant function

Write the cosecant function in terms of ratio of lengths of the sides of right angled triangle.

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

Express it in Reciprocal form

Write the ratio of sides of cosecant function in reciprocal form.

$\csc{\theta} \,=\, \dfrac{1}{\dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}}$

Expressing it in terms of Sine

Actually, the ratio of length of the opposite side to length of the hypotenuse is called sine function as per trigonometry.

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

Therefore, it is proved that cosecant function can be expressed in terms of sine function as per this law.

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

The reciprocal relation of sine with cosecant and cosecant with sine functions are used as formulas to replace sine by cosecant and vice-versa in trigonometric mathematics.



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