Math Doubts

Sine

The relation between angle and ratio of lengths of opposite side to hypotenuse in a right triangle is called sine.

Introduction

Sine is a name and it is used to represent the ratio of lengths of opposite side to hypotenuse at a particular angle in a right triangle. It is written in ratio form and also written as sine with angle alternatively.

1. Ratio

The value of sine at an angle is calculated by the ratio of lengths of opposite side to hypotenuse.

$\dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$

So, sine is called as a trigonometric ratio generally.

2. Function

Alternatively, the value of sine at an angle is expressed mathematically by writing sine in its short form $\sin$ and then corresponding angle of the right triangle.

For example, if angle of right triangle is denoted by $x$, then sine of angle $x$ is written as $\sin{x}$ in trigonometry. $\sin{x}$ is a function form. So, it is usually called as sin function in mathematics.

Thus, sin functions like $\sin{A}$, $\sin{\alpha}$, $\sin{\beta}$, and etc. are defined in mathematics.

Mathematical form

$\Delta BAC$ is a right angled triangle and its angle is theta ($\theta$).

right angled triangle

sine of angle is written as $\sin{\theta}$ in this case.

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

It is used as a formula to calculate the value of sine at any angle of the right triangle.

In this example, $BC$ is length of opposite side (perpendicular) and $AC$ is length of hypotenuse.

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

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