$\sin{45^\circ}$ value

Exact value

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

Introduction

The value of sine in a $45$ degrees right triangle (or right angled triangle) is called the sine of angle forty five degrees. In sexagesimal system, the angle forty five degrees is written as $45^\circ$. So, the sine of angle forty five degrees is written as $\sin{45^\circ}$ in trigonometry. Now, let’s learn what the sin $45$ degrees value is.

Fraction form

The exact value of sin $45$ degrees in fraction form is written as a fraction in radical form, and it is equal to $1$ divided by square root of $2$.

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

Decimal form

The trigonometric function sine for a standard angle $45$ degrees value is an irrational number. So, it can also be evaluated in decimal form as follows.

$\sin{(45^\circ)}$ $\,=\,$ $0.7071067812\ldots$

$\implies$ $\sin{(45^\circ)}$ $\,\approx\,$ $0.7071$

Other forms

The sine of $45$ degrees can be written in trigonometry alternatively in two different forms.

Circular system

The sine of angle $45$ degrees is written as sine of pi divided by four radians in circular system. So, the sine of $\pi$ divided by $4$ radians is equal to $1$ divided by square root of $2$ in fraction form and its value in decimal form is equal to $0.7071$ approximately.

$\sin{\Big(\dfrac{\pi}{4}\Big)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,\approx\,$ $0.7071$

Centesimal system

Similarly, the sine of $45$ degrees is also written as sine of $50$ gradians as per centesimal system. Therefore, the exact value of sine of fifty grades is equal to one divided by square root of two in fraction form and sine of $50$ grads is approximately equal to $0.7071$ in decimal form.

$\sin{(50^g)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,\approx\,$ $0.7071$

Proof

Learn how to find the exact value of sine $45$ degrees as an irrational fraction one divided by square root of two by constructing a right triangle with an angle forty five degrees.