Math Doubts

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

The value of sin of $30$ degrees can be derived in theoretical approach but it requires knowledge on properties of right triangle when its angle is equal to $30^°$. You can even find the value of sin of angle $30$ degrees geometrically by constructing a right triangle with $30$ degrees angle with the help of geometric tools.

construction of right triangle whose angle equals to 30 degrees for finding the value of sin30 degrees
  1. From point $\small G$, draw a straight line on a plane horizontally.
  2. Draw a $30$ degrees line from point $\small G$ by using protractor and ruler.
  3. Set compass to any length. It is set to $7 \, cm$ in this case by considering measurement of ruler. Later, draw an arc on $30$ degrees line from point $\small G$ and it intersects the $30$ degrees line at point $\small H$.
  4. Finally, draw a perpendicular line to horizontal line from point $\small H$ and it intersects the horizontal line perpendicularly at point $\small I$. This procedure constructs a right triangle (called $\small \Delta IGH$) geometrically.

In $\Delta IGH$, the length of hypotenuse is $GH = 7 \, cm$ and the angle of right triangle is $30$ degrees. So, write sin of angle $30$ degrees in the form of ratio of lengths of the sides.

measuring opposite side when angle of right triangle is 30 deg

$\sin{(30^°)} \,=\, \dfrac{HI}{GH}$

The length of opposite side is required to calculate the exact value of sin $30$ degrees. So, take a ruler and measure the length of the opposite side. You will measure that the length of opposite side is $3.5 \, cm$ exactly.

$\implies \sin{(30^°)} \,=\, \dfrac{3.5}{7}$

$\implies \require{cancel} \sin{(30^°)} \,=\, \dfrac{\cancel{3.5}}{\cancel{7}}$

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

It is exact value of sin of $30$ degrees in fraction and its value in decimal is given below.

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

In this way, the value of sine of angle $30$ degrees can be derived experimentally in trigonometry by using geometric tools.

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