Math Doubts

sin 0°

sine of angle 0 degrees is 0


The value of sine when the angle of the right angled triangle equals to $0^\circ$ is called $\sin 0^\circ$.

The exact value of $\sin 0^\circ$ is obtained mathematically by calculating the ratio of lengths of opposite side to hypotenuse when the angle of the right angled triangle is $0^\circ$.


The value of $\sin 0^\circ$ is derived in mathematics by using two fundamental geometrical approaches.


Fundamental approach

According to properties of the right angled triangle, when the angle of the right angled triangle is $0^\circ$, the length of the opposite side is zero but the length of the hypotenuse is not zero.

properties of right angled triangle when angle is 0 degrees

$Length \, of \, Opposite \, side = 0$

Calculate the ratio of length of opposite side to length of hypotenuse to obtain value of sine at $0^\circ$.

$$\frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse} = \frac{0}{d}$$

The ratio of length of opposite side to length of hypotenuse at angle $0^\circ$ is known as $\sin 0^\circ$.

$$\sin 0^\circ = \frac{0}{d}$$

$\implies \sin 0^\circ = 0$


Practical approach

Geometrically, the value of $\sin 0^\circ$ can be calculated by constructing a right angled triangle using geometric tools.

right angled triangle with 10 centimeters length and 0 degrees angle
  1. Take a ruler and draw a straight line of any length in horizontal direction. The left side point of it is called as point $G$.
  2. Take protractor and coincide its centre with point $G$ and also coincide the right side base line with horizontal line. Identify $0^\circ$ angle by using bottom scale but the horizontal line direction is exact $0^\circ$ angle. So, there is no use of protractor in this case.
  3. Take ruler and compass, then set the distance between points of pencil’s lead and needle to $10$ centimetres. Later, draw an arc from point $G$ on $0$ degrees angle line and it cuts the line at point $H$.
  4. A perpendicular line should be drawn from point $H$ to horizontal line but it’s not possible to do it in this case because the $0^\circ$ angle line and horizontal line are at same position on the plane. Therefore, assume a perpendicular straight line is drawn from point $H$ to horizontal line and it meets the horizontal line at point $I$.

Right angled triangle ($\Delta HGI$) is constructed according to this geometrical procedure.

In right angled triangle $HGI$, the points $H$ and $I$ are at same location. So, the distance between them is zero. Therefore, the length of the opposite side ($HI$) is zero. The length of the hypotenuse ($GH$) is $10$ centimeters. The angle of the right angled triangle is $0^\circ$.

Now, calculate the value of $\sin 0^\circ$ by using this information.

$$\sin 0^\circ = \frac{HI}{GH}$$

$$\implies \sin 0^\circ = \frac{0}{10}$$

$\implies \sin 0^\circ = 0$


The fundamental geometrical approach and geometrical approach by using geometrical tools have given same result.

$\therefore \,\, \sin 0^\circ = 0$


It is expressed in mathematics in three possible forms as per different angle measuring systems.

In sexagesimal system, it is written as.

$\sin 0^° = 0$

In circular system, it is also written as.

$\sin 0 = 0$

In centesimal system, it is also expressed as.

$\sin 0^g = 0$

Follow us
Email subscription
Math Doubts
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
Follow us on Social Media
Mobile App for Android users Math Doubts Android App
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more