$\sin{(45^\circ)}$ value

$\sin{(45^\circ)} \,=\, \dfrac{1}{\sqrt{2}}$

The value of sine in a forty five degrees right triangle is called the sine of angle forty five degrees.

Introduction

The value of sin function when angle of right triangle equals to $45^\circ$ is called sine of angle $45$ degrees. It is written as $\sin{(45^\circ)}$ mathematically in sexagesimal system.

The exact value of sin of $45$ degrees in fraction is $\dfrac{1}{\sqrt{2}}$. It is an irrational number and is equal to $0.7071067812\ldots$ in decimal form. The value of sin of angle $45$ degrees is considered as $0.7071$ approximately in mathematics. The value of $\sin{(45^\circ)}$ is generally called as trigonometric function or trigonometric ratio of standard angle.

Alternative form

$\sin{(45^\circ)}$ is alternatively written as $\sin{\Big(\dfrac{\pi}{4}\Big)}$ in circular system and also written as $\sin{(50^g)}$ in centesimal system.

$(1) \,\,\,$ $\sin{\Big(\dfrac{\pi}{4}\Big)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,=\,$ $0.7071067812\ldots$

$(2) \,\,\,$ $\sin{(50^g)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,=\,$ $0.7071067812\ldots$

Proof

You have learnt the exact value of sin of $45$ degrees in both fraction and decimal form. Now, it is time to learn how to derive the value of $\sin{\Big(\dfrac{\pi}{4}\Big)}$ in trigonometry mathematically.