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.

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.

List of most recently solved mathematics problems.

Jul 04, 2018

Limit (Calculus)

Evaluate $\displaystyle \large \lim_{x \,\to\, \tan^{-1}{3}} \normalsize {\dfrac{\tan^2{x}-2\tan{x}-3}{\tan^2{x}-4\tan{x}+3}}$

Jun 23, 2018

Limit (Calculus)

Evaluate $\displaystyle \large \lim_{x \to 0} \normalsize \dfrac{e^{x^2}-\cos{x}}{x^2}$

Jun 22, 2018

Integral Calculus

Evaluate $\displaystyle \int \dfrac{1+\cos{4x}}{\cot{x}-\tan{x}} dx$

Jun 21, 2018

Limit

Evaluate $\displaystyle \large \lim_{x \to \infty} \normalsize {\sqrt{x^2+x+1}-\sqrt{x^2+1}}$

Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers.
Know more

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.