Math Doubts

Even or Odd identity of Sine function

Formula

$\sin{(-\theta)} \,=\, -\sin{\theta}$

The sine of negative angle is equal to minus sine of angle, is called the even or odd trigonometric identity of sine function

Introduction

Sine functions are appeared with negative angles in some cases in mathematics but we only know how to evaluate the sine of positive angles in trigonometry. Actually, the sine of negative angle can be evaluated from the sine of positive angle by the relation between sines of positive and negative angles.

Let theta represents an angle of a right triangle (or right angled triangle), the sine of angle theta is written as $\sin{\theta}$ and the sine of angle negative theta is written as $\sin{(-\theta)}$ in trigonometric mathematics.

The sine of negative angle is mathematically equal to the negative sine of angle.

$\sin{(-\theta)} \,=\, -\sin{\theta}$

It clears that the sine function is an odd function. This mathematical equation is used as a formula in mathematics and it is called in the following two ways.

  1. Even or Odd identity of Sine function
  2. Negative angle identity of Sine function

Usage

The sine even odd trigonometric identity is used in two cases in mathematics.

Positive to Negative

It is used to transform sine of angle in the form of sine of negative angle.

$\implies$ $\sin{\theta} \,=\, -\sin{(-\theta)}$

Negative to Positive

It is also used to transform sine of negative angle in the form of sine of angle.

$\implies$ $\sin{(-\theta)} \,=\, -\sin{\theta}$

Example

Evaluate $\sin{(-60^°)}$

The sine negative angle identity can be used to evaluate the sine of minus sixty degrees.

$\implies$ $\sin{(-60^°)}$ $\,=\,$ $-\sin{(60^°)}$

Now, substitute the sine of angle sixty degrees to evaluate the sine of minus sixty degrees value.

$\implies$ $\sin{(-60^°)}$ $\,=\,$ $-\dfrac{\sqrt{3}}{2}$

Proof

Learn the geometric proof to learn how to derive the sine of even or odd identity in trigonometric mathematics.

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