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

## Exact value

$\cos{(54^\circ)} \,=\, \dfrac{\sqrt{10-2\sqrt{5}}}{4}$

### Introduction

The value of cosine in a fifty four degrees right triangle is called the cosine of angle fifty four degrees.

In Sexagesimal angle measuring system, the angle fifty four degrees is written as $54^\circ$ and the cosine of fifty four degrees is written as $\cos{54^\circ}$. Now, let us learn what the cos $54$ degrees is in trigonometry.

#### Fraction form

The exact value of cos $54$ degrees is an irrational number and it is written in fraction form as the square root of ten minus two times square root of five divided by four.

$\cos{(54^\circ)} \,=\, \dfrac{\sqrt{10-2\sqrt{5}}}{4}$

#### Decimal form

The cos $54$ degrees value can be expressed in decimal form as an infinitely extended number but its exact value is approximately considered as follows.

$\cos{(54^\circ)}$ $\,=\,$ $0.5877852522\cdots$

$\implies$ $\cos{(54^\circ)}$ $\,\approx\,$ $0.5878$

#### Other forms

In trigonometry, the cos $54$ degrees is also written in two other mathematical forms alternatively.

##### Circular system

The cos of $45$ degrees is written as the cosine of three times pi divided by ten radians in circular angle measuring system. So, the sine $3\pi/10$ radians value in surd form is exactly equal to the square root of $10$ minus $2$ times $\sqrt{5}$ divided by $4$ and its value in decimal form is $0.5878$ approximately.

$\cos{\Big(\dfrac{3\pi}{10}\Big)}$ $\,=\,$ $\dfrac{\sqrt{10-2\sqrt{5}}}{4}$ $\,\approx\,$ $0.5878$

##### Centesimal system

In Centesimal system, the cos of fifty four degrees is written as cosine of $60$ grades and the exact value of cos $60$ gradians in fraction is equal to the square root of $10$ minus two times $\sqrt{5}$ divided by $4$, and the cos $60$ grads value in decimal form is equal to $0.5878$ approximately.

$\cos{(60^g)}$ $\,=\,$ $\dfrac{\sqrt{10-2\sqrt{5}}}{4}$ $\,\approx\,$ $0.5878$

#### Proof

Learn how to find the exact value of cosine of angle fifty four degrees as the square root of ten minus two times square root of five divided by four.

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