# $\sin{(36^\circ)}$ value

$\sin{(36^\circ)} \,=\, \dfrac{\sqrt{10-2\sqrt{5}}}{4}$

The value of sine in a thirty six degrees right triangle is called the sine of angle thirty six degrees.

## Introduction

The sine of angle thirty six degrees is a value that expresses the ratio of length of opposite side to length of hypotenuse when the angle of a right triangle is thirty six degrees.

In the sexagesimal system, the sine of angle thirty six degrees is written as $\sin{(36^\circ)}$ mathematically and its exact value in fraction form is the quotient of square root of ten minus two times the square root of five by four. In trigonometry, it is written in mathematical form as follows.

$\sin{(36^\circ)} \,=\, \dfrac{\sqrt{10-2\sqrt{5}}}{4}$

The value of sine of thirty six degrees is an irrational number and its value can be written in the following decimal form.

$\implies$ $\sin{(36^\circ)} \,=\, 0.5877852522\cdots$

$\implies$ $\sin{(36^\circ)} \,\approx\, 0.5878$

The sine of thirty six degrees can also be written in two other mathematical forms.

### Circular system

In the circular system, the sine of thirty six degrees is written as the sin of quotient of pi by five radian. It is written in mathematical form as $\sin{\Big(\dfrac{\pi}{5}\Big)}$.

$\sin{\Big(\dfrac{\pi}{5}\Big)} \,=\, \dfrac{\sqrt{10-2\sqrt{5}}}{4}$

### Centesimal system

Similarly, the sine thirty six degrees is written as sine of angle forty grades and it is written as $\sin{\Big(40^g\Big)}$ in mathematical form in the centesimal system.

$\sin{\Big(40^g\Big)} \,=\, \dfrac{\sqrt{10-2\sqrt{5}}}{4}$

#### Proofs

Learn how to prove the exact value of sine of thirty six degrees in trigonometric and geometric methods.

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