According to even-odd identity of sine function, the sine of negative angle is equal to negative sign of sine of angle.

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

This negative angle trigonometric identity of sine function can be proved geometrically in mathematical form.

A line segment is rotated in anticlockwise direction from zero angle position to make an angle with horizontal axis. The line segment is mathematically expressed as $\overline{OA}$ at its final position.

The angle made by the line segment $OA$ with horizontal axis is denoted by a symbol theta.

A perpendicular line to positive $x$ axis is drawn from point $A$ and it intersects the axis perpendicularly at point $B$. Thus, a right triangle $AOB$ is constructed geometrically and it is written as $\Delta AOB$ in mathematics.

In the first quadrant, the divisions of both coordinate axes are represented by positive values. Hence, the lengths of adjacent side and opposite side are denoted by $x$ and $y$ algebraically.

According to the $\Delta AOB$, the sine of angle theta can be written mathematically in ratio form.

$\sin{\theta} \,=\, \dfrac{AB}{OA}$

$\implies$ $\sin{\theta} \,=\, \dfrac{y}{OA}$

Actually, the length of hypotenuse is unknown but it can be evaluated from the lengths of opposite and adjacent sides by the Pythagorean Theorem.

$OA$ $\,=\,$ $\sqrt{OB^2+AB^2}$ $\,=\,$ $\sqrt{x^2+y^2}$

$\implies$ $\sin{\theta} \,=\, \dfrac{y}{\sqrt{x^2+y^2}}$

A line segment is rotated in clockwise direction from zero angle position to make same angle with horizontal axis. The line segment is written mathematically as $\overline{OC}$ at its final position.

The angle made by the line segment $OC$ with horizontal axis is denoted by same symbol theta but the line is rotated in clockwise direction. Hence, the angle is written as negative theta ($-\theta$) as per sense of angle system.

From point $C$, a perpendicular line is drawn to horizontal axis and it intersects the $x$ axis perpendicularly at point $D$. In this way, a right angled triangle (expressed as $\Delta COD$) is constructed geometrically.

In the fourth quadrant, each division of $x$ axis is denoted by a positive value but each division of $y$ axis is denoted by a negative value. Therefore, the lengths of adjacent side and opposite side are represented in algebraic form as $x$ and $-y$ respectively.

According to the $\Delta COD$, the sine of angle negative theta can be written in ratio form in mathematical form.

$\sin{(-\theta)} \,=\, \dfrac{CD}{OC}$

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

The length of hypotenuse can be evaluated from the lengths of opposite side and adjacent side by the Pythagorean Theorem.

$OA$ $\,=\,$ $\sqrt{OC^2+CD^2}$ $\,=\,$ $\sqrt{x^2+(-y)^2}$

$\implies$ $OA$ $\,=\,$ $\sqrt{OC^2+CD^2}$ $\,=\,$ $\sqrt{x^2+y^2}$

$\implies$ $\sin{(-\theta)} \,=\, \dfrac{-y}{\sqrt{x^2+y^2}}$

According to the above two steps, the sine functions are expressed in ratio form of the sides of the two triangles.

$\sin{\theta} \,=\, \dfrac{y}{\sqrt{x^2+y^2}}$

$\sin{(-\theta)} \,=\, -\dfrac{y}{\sqrt{x^2+y^2}}$

The value of sine of angle theta is also in the value of the sine of negative angle theta but it is with negative symbol.

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

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

Geometrically, it is proved that the sine of negative angle is equal to the negative sign of sine of angle. It is called as sine of negative angle identity or even-odd identity of sine function in trigonometry mathematics.

Latest Math Topics

Jan 06, 2023

Jan 03, 2023

Jan 01, 2023

Dec 26, 2022

Dec 08, 2022

Latest Math Problems

Jan 31, 2023

Nov 25, 2022

Nov 02, 2022

Oct 26, 2022

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the maths problems in different methods with understandable steps.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved