Math Doubts

$\sin{(18^°)}$ value

$\sin{(18^\circ)}$ $\,=\,$ $\dfrac{\sqrt{5}-1}{4}$

The value of sine in an eighteen degrees right triangle is called the sine of angle eighteen degrees.

Introduction

The sine of angle eighteen degrees is a value that represents the ratio of length of opposite side to length of hypotenuse when the angle of a right angled triangle is eighteen degrees.

In sexagesimal system, the sine of angle eighteen degrees is written as $\sin{(18^\circ)}$ mathematically in trigonometry and the exact value of sine of angle eighteen degrees in fraction form is quotient of square root of five minus one by four. So, it can be written in the following mathematical form.

$\sin{(18^\circ)}$ $\,=\,$ $\dfrac{\sqrt{5}-1}{4}$

The exact value of sine of eighteen degrees is completely an irrational number and its value can be written in decimal form as follows.

$\implies$ $\sin{(18^\circ)}$ $\,=\,$ $0.3090169943\ldots$

$\implies$ $\sin{(18^\circ)}$ $\,\approx\,$ $0.309$

The sin of eighteen degrees is also expressed in two other forms.

circular system

The sine eighteen degrees is expressed as the sin of quotient of pi by ten radian in the circular system. Therefore, it can be written in mathematical form as $\sin{\Big(\dfrac{\pi}{10}\Big)}$.

$\sin{\Big(\dfrac{\pi}{10}\Big)} \,=\, \dfrac{\sqrt{5}-1}{4}$

Centesimal system

According to the centesimal system, the sine of angle eighteen degrees is written as sine of angle twenty grades. So, it is written as $\sin{\Big(20^g\Big)}$ in mathematical form.

$\sin{\Big(20^g\Big)} \,=\, \dfrac{\sqrt{5}-1}{4}$

Proofs

The sine of eighteen degrees value can be derived exactly in two different methods in mathematics.

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