Math Doubts

$\sin{18^\circ}$ value

Exact 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.

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.

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


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

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