Math Doubts

$\sin{(0^°)}$ Proof

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.

Theoretical approach

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.

properties of right triangle whose angle equals to 0 degrees

$\Delta MON$ 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.

Practical approach

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.

construction of right triangle with 0 degrees for sin 0 degrees value
  1. Draw a straight line horizontally from point $A$ on the plane.
  2. 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.
  3. Now, draw a straight line from point $A$ through the $0^°$ angle point by ruler but the line lies on the horizontal line.
  4. 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 $B$.
  5. 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 $BAC$.

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

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

Measuring opposite side of right triangle when angle is 0 degrees

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.

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