# $\sin{54^\circ}$ value

## Exact value

$\sin{(54^\circ)} \,=\, \dfrac{\sqrt{5}+1}{4}$

### Introduction

The value of sine in a $54$ degrees right triangle is called the sine of angle fifty four degrees. The fifty four degrees is written as $54^\circ$ as per Sexagesimal system and the sine of fifty four degrees is written as $\sin{54^\circ}$ mathematically in trigonometry. It is time to learn what the sin $54$ degrees value is.

#### Fraction form

The trigonometric function sine $54$ degrees value is a fraction in irrational form and its exact value is equal to the square root of $5$ plus $1$ divided by $4$.

$\sin{54^\circ}$ $\,=\,$ $\dfrac{\sqrt{5}+1}{4}$

#### Decimal form

The exact value of sin $54$ degrees is expressed as a fraction in radical form and it can be expressed in decimal form as follows.

$\sin{(54^\circ)}$ $\,=\,$ $0.8090169943\ldots$

$\implies$ $\sin{(54^\circ)}$ $\,\approx\,$ $0.809$

#### Other forms

Let’s learn how to write sine of angle fifty four degrees alternatively in two other forms.

##### Circular system

According to the circular system, the sine of $54$ degrees is written as sine of three times pi divided by ten radians, and the exact value of the sin $3\pi$ divided by $10$ in surd form is $\sqrt{5}$ plus $1$ divided by $10$. Similarly, its exact value in decimal form is approximately equal to $0.809$.

$\sin{\bigg(\dfrac{3\pi}{10}\bigg)}$ $\,=\,$ $\dfrac{\sqrt{5}+1}{4}$ $\,\approx\,$ $0.809$

##### Centesimal system

Likewise, the sine of angle $54$ degrees is also written as the sine of sixty gradians in the centesimal system. So, the exact value of sine $60$ grades in irrational fraction form is equal to the square root of five plus one divided by ten and the approximate value of sine $60$ grads in decimal form is $0.809$.

$\sin{(60^g)}$ $\,=\,$ $\,=\,$ $\dfrac{\sqrt{5}+1}{4}$ $\,\approx\,$ $0.809$

##### Proof

Let’s learn how to find the exact value of sin $54$ degrees in both trigonometric and geometric methods.

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