Math Doubts

$\cos{(18^°)}$ value

$\cos{(18^°)}$ $\,=\,$ $\dfrac{\sqrt{10+2\sqrt{5}}}{4}$

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

Introduction

The cosine of angle eighteen degrees is a value that represents the ratio of length of adjacent side to length of hypotenuse when the angle of a right triangle (or right angled triangle) is eighteen degrees.

The cosine of angle eighteen degrees is written as $\cos{(18^°)}$ in trigonometry as per the sexagesimal system and its exact value in fraction form is quotient of square root of ten plus two times square root of five by four. Hence, it is written mathematically in the following form.

$\cos{(18^°)}$ $\,=\,$ $\dfrac{\sqrt{10+2\sqrt{5}}}{4}$

The cos of eighteen degrees value is exactly an irrational number and its value can be written in decimal form too as follows.

$\implies$ $\cos{(18^°)}$ $\,=\,$ $0.9510565162\ldots$

$\implies$ $\cos{(18^°)}$ $\,\approx\,$ $0.9511$

The cosine eighteen degrees is also written in two other forms in trigonometric mathematics.

circular system

The cos of eighteen degrees can be written as the cos of quotient of pi by ten radian as per the circular system. So, it is written in mathematical form as $\sin{\Big(\dfrac{\pi}{10}\Big)}$.

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

Centesimal system

On the basis of the centesimal system, the cosine of angle eighteen degrees is written as cos of angle twenty grades. Therefore, it is written as $\cos{\Big(20^g\Big)}$ in mathematical form.

$\cos{\Big(20^g\Big)}$ $\,=\,$ $\dfrac{\sqrt{10+2\sqrt{5}}}{4}$

Proofs

The exact value of cosine of eighteen degrees can be derived in two distinct methods in mathematics.

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