Math Doubts

Proof of $\sin{(30^°)}$ in Theoretical Geometric method

There is a special geometrical relation between opposite side and hypotenuse when angle of right triangle is $30$ degrees and this property is used to derive the exact value of $\sin{30^°}$ in fraction and decimal form mathematically.

Theoretically, the length of opposite side is always half of the length of hypotenuse when angle of right triangle is $30$ degrees.

right triangle with 30 degrees angle for finding sin30 value

In $\Delta POQ$, the length of hypotenuse is taken as $d$. So, the length of the opposite side should be half of it. Hence, the length of opposite side is $\dfrac{d}{2}$.

Calculate the ratio of length of opposite side to length of hypotenuse.

$\dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$ $\,=\,$ $\dfrac{PQ}{OP}$

$\implies$ $\dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$ $\,=\,$ $\dfrac{\dfrac{d}{2}}{d}$

$\implies$ $\dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$ $\,=\,$ $\dfrac{d}{2 \times d}$

$\implies$ $\dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$ $\,=\,$ $\require{cancel} \dfrac{\cancel{d}}{2 \times \cancel{d}}$

$\implies$ $\dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$ $\,=\,$ $\dfrac{1}{2 \times 1}$

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

The ratio of lengths of opposite side to hypotenuse is called sine in trigonometry and the ratio between them is calculated when angle of right triangle $POQ$ is $30^°$. Trigonometrically, the ratio of lengths of opposite side to hypotenuse when angle of right triangle equals to $30^°$ is called sin of angle $30$ degrees.

$\,\,\, \therefore \,\,\,\,\,\,$ $\sin{(30^°)}$ $\,=\,$ $\dfrac{1}{2}$

It is exact value of sin $30$ degrees in fraction form and it can be written in decimal as well.

$\,\,\, \therefore \,\,\,\,\,\,$ $\sin{(30^°)}$ $\,=\,$ $0.5$

Thus, the value of sine of angle $30$ degrees is derived theoretically in geometrical system.



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