$\sin{18^\circ}$ value

Exact value

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

Introduction

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

In sexagesimal angle measuring system, the angle eighteen degrees is written as $18^\circ$ in mathematics and the sine of $18$ degrees is expressed as $\sin{18^\circ}$ in trigonometry. Let’s know what the sin $18$ degrees value is.

Fraction form

The sin $18$ degrees value is exactly equal to the square root of five minus one divided by four.

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

Decimal form

The exact value of sin of $18$ degrees is a fraction in radical form. However, the surd in fraction can be evaluated in decimal form to find the sine of angle $18$ degrees. It is an irrational number, which means it is a number with infinitely extended digits. For that reason, the exact value of sin $18$ degrees is approximately considered in decimal form.

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

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

Other forms

The sine of eighteen degrees is alternatively written in two different forms in trigonometry.

Circular system

The sine of $18$ degrees is written as sine of pi divided by ten radians in circular angle measuring system. So, the sin $\pi$ divided by $10$ radians in fraction form is equal to $\sqrt{5}$ minus $1$ divided by $4$ and its approximate value in decimal form is $0.309$.

$\sin{\Big(\dfrac{\pi}{10}\Big)}$ $\,=\,$ $\dfrac{\sqrt{5}-1}{4}$ $\,\approx\,$ $0.309$

Centesimal system

According to the Centesimal system, the sine of angle $18$ degrees is written as sine of angle twenty gradians. Therefore, the exact value of sin of $20$ grades is equal to square root of $5$ minus $1$ divided by $4$ and its value in decimal form is $0.309$ approximately.

$\sin{\big(20^g\big)}$ $\,=\,$ $\dfrac{\sqrt{5}-1}{4}$ $\,\approx\,$ $0.309$

Proofs

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