The exact value of sin of $0$ degrees can be derived in trigonometry by the geometry in two different approaches. Now, it is your time to know how to derive the $\sin{(0)}$ value in trigonometry mathematically.

First of all, you must have to know the properties of right triangle when angle of right triangle equals to $0$ degrees. Actually, the length of opposite side is zero if angle of right triangle is zero degrees. On the basis of this property, the value of $\sin{(0^g)}$ is derived theoretically in trigonometric mathematics.

$\Delta MON$ is a triangle which represents a right triangle when its angle is zero angle. In this case, the length of opposite side ($MN$) is zero but the length of hypotenuse is taken as $d$. Now, express sin of zero degrees in its ratio form.

$\dfrac{MN}{ON} \,=\, \dfrac{0}{d}$

$\implies \dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse} = 0$

The ratio of lengths of opposite side to hypotenuse is calculated when angle of right triangle is $0$ degrees. So, the value is denoted by $\sin{(0^°)}$ in trigonometry.

$\,\,\, \therefore \,\,\,\,\,\, \sin{(0^°)} \,=\, 0$

Thus, it is proved that the exact value of sin $0$ degrees is zero by the theoretical geometric method.

The value of $\sin{(0^g)}$ can also be derived geometrically in practical approach by constructing a right triangle with zero angle by using geometrical tools.

- Draw a straight line horizontally from point $A$ on the plane.
- Coincide the point $A$ with centre of protractor and also coincide right side base line of protractor with horizontal line. Now, mark a point at $0$ degrees angle. It obviously lies on the same horizontal line.
- Now, draw a straight line from point $A$ through the $0^°$ angle point by ruler but the line lies on the horizontal line.
- Set your compass to any length by ruler. In this example, it is set to $5 \, cm$ and then draw an arc on zero angle line from point $A$ and take the intersecting point as point $B$.
- We have to draw a perpendicular line to horizontal line from point $B$ but it is not possible in this case because the horizontal line and $0$ angle line both are drawn one on one. However, just imagine that a perpendicular line is drawn to horizontal line from point $B$ and it intersects the horizontal line at point $C$. Thus, a right triangle is constructed with $0$ angle and it is written as $\Delta BAC$.

It is time to find the value of $\sin{(0)}$ by calculating the ratio of lengths of opposite side to hypotenuse of the right triangle $BAC$.

$\sin{(0^°)} = \dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$

$\implies \sin{(0^°)} \,=\, \dfrac{BC}{AB}$

The length of opposite side ($BC$) is zero but the length of hypotenuse ($AB$) is $5 \, cm$ in $\Delta BAC$. Now, calculate the sin of $0$ degrees by this information.

$\implies \sin{(0^°)} \,=\, \dfrac{BC}{AB} = \dfrac{0}{5}$

$\,\,\, \therefore \,\,\,\,\,\, \sin{(0^°)} \,=\, 0$

Therefore, it is also proved that the value of sin of zero radian is also equal to zero in the practical geometric approach.

Compare the values of $\sin{(0^°)}$, derived from theoretical and practical geometric approaches. They both are same and it is equal to zero.

Latest Math Topics

Latest Math Problems

Email subscription

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.