Math Doubts

$\sin{(18^°)}$ value

The value of sine function when angle of right triangle equals to $18^°$ is called sin of angle $18$ degrees. In mathematics, it is written as $\sin{(18^°)}$ as per sexagesimal system.

$\sin{(18^°)} \,=\, \dfrac{\sqrt{5}-1}{4}$

It is an exact value of sin of $18$ degrees in fraction. Actually, it is an irrational number and equals to $0.3090169943\ldots$ in decimal form. In mathematics, the approximate value of sin of angle $18$ degrees is considered as $0.30902$.

Alternative form

The $\sin{(18^°)}$ is expressed alternatively as $\sin{\Big(\dfrac{\pi}{18}\Big)}$ in circular system and also expressed as $\sin{(20^g)}$ in centesimal system.

$(1) \,\,\,$ $\sin{\Big(\dfrac{\pi}{18}\Big)}$ $\,=\,$ $\dfrac{\sqrt{5}-1}{4}$ $\,=\,$ $0.3090169943\ldots$

$(2) \,\,\,$ $\sin{(20^g)}$ $\,=\,$ $\dfrac{\sqrt{5}-1}{4}$ $\,=\,$ $0.3090169943\ldots$

Proof

Now, you know the exact value of $\sin{\Big(\dfrac{\pi}{18}\Big)}$ and it is time to learn how to derive the exact value of sin $18$ degrees in trigonometry.



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