The value of sine when the angle of right angled triangle equals to $45$ degrees is called $\sin 45^\circ$.

The value of $\sin 45^\circ$ is calculated by calculating the ratio of length of opposite side to length of hypotenuse when the angle of the right angled triangle is equal to $45^\circ$.

The value of $\sin 45^\circ$ can be calculated in two different methods.

1

$\Delta RPQ$ is a right angled triangle and the angle of the triangle is $45^\circ$. According to properties of triangle, the lengths of opposite and adjacent sides are equal if angle of the right angled triangle is $45^\circ$.

So, assume the length of opposite side and length of adjacent side is $l$.

Apply Pythagorean Theorem to this triangle.

$r^2 = l^2 + l^2$

$\implies r^2 = 2l^2$

$\implies r = \sqrt{2}.l$

$$\implies \frac{l}{r} = \frac{1}{\sqrt{2}}$$

$l$ and $r$ are lengths of opposite side and hypotenuse, and they are in ratio form.

$$\implies \frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse} = \frac{1}{\sqrt{2}}$$

According to definition of sine function, the ratio of length of opposite side to length of hypotenuse is sine and the angle of this right angled triangle is $45^\circ$. So, the ratio of length of opposite side to length of hypotenuse is $\sin 45^\circ$.

$$\therefore \, \sin 45^\circ = \frac{1}{\sqrt{2}}$$

$$\sin 45^\circ = \frac{1}{\sqrt{2}} = 0.707106781…$$

It is the absolute value of sine of angle $45^\circ$.

2

Geometrically, the value of sine of angle $45^\circ$ can be derived by drawing a line of any length with $45^\circ$ angle to construct a right angled triangle.

- Draw a horizontal line of any length and call left side point as Point $F$.
- Take Protractor, coincide point $F$ with centre of the Protractor and also coincide the horizontal line with right side base line of protractor. Then mark a point at $45$ degrees.
- Take scale and draw a line from point $F$ through $45^\circ$ angle marked point.
- Take compass and set the length to $7$ centimetres with the help of a scale. Then draw an arc on $45^\circ$ angle line from point $F$. The arc cuts the $45$ degrees line at point $G$.
- Take set square and draw a line from point $G$ to horizontal line but it should be perpendicular to the horizontal line. The line from point $G$ touches the horizontal line at point $H$.

The geometrical process formed a right angled triangle and it is $\Delta GFH$. Apply definition of sine of angle to this triangle.

$$\sin 45^\circ = \frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$$

$$\implies \sin 45^\circ = \frac{GH}{FG}$$

Length of the hypotenuse is $7$ centimeters but length of the opposite side is unknown. So, take scale and measure the length of $GH$. It will be $4.95$ centimetres. Substitute these two values to get the value of sine of $45$ degrees.

$$\implies \sin 45^\circ = \frac{GH}{FG} = \frac{4.95}{7}$$

$\implies \sin 45^\circ = 0.707142857…$

It is expressed in three different forms in mathematics.

In sexagesimal system, it is usually written in mathematics as follows.

$$\sin 45^° = \frac{1}{\sqrt{2}}$$

In circular system, it is expressed in mathematics as follows.

$$\sin \Bigg(\frac{\pi}{4}\Bigg) = \frac{1}{\sqrt{2}}$$

In centesimal system, it is written in mathematical form as follows.

$$\sin 50^g = \frac{1}{\sqrt{2}}$$