Proofs of $\sin{(36^\circ)}$ value

The exact value of sin 36 degrees in fraction is equal to the quotient of square root of ten minus two times square root of five by four.

$\sin{(36^\circ)}$ $\,=\,$ $\dfrac{\sqrt{10-2\sqrt{5}}}{4}$

The sine of angle pi by five radian can be derived in two distinct methods in mathematics. Therefore, let’s learn both methods to derive the sine of angle forty grades value mathematically.

Trigonometric method

Learn how to prove the exact value of sine of angle thirty six degrees in fraction form in trigonometric approach.

Geometric method

Learn how to derive the value of sine of angle pi by five in geometric method.

Conclusion

As per the trigonometric method, the sine of angle forty grades is derived in fraction form as follows.

$\sin{(36^\circ)}$ $\,=\,$ $\dfrac{\sqrt{10-2\sqrt{5}}}{4}$

$\implies$ $\sin{(36^\circ)}$ $\,=\,$ $0.5877852522\cdots$

$\implies$ $\sin{(36^\circ)}$ $\,\approx\,\,$ $0.5878$

As per the geometrical approach, the sine of angle pi by five radian is derived in decimal form as follows.

$\sin{(36^\circ)}$ $\,\approx\,$ $0.5946$

The sine of thirty six degrees is $0.5878$ in trigonometric method but its value is $0.5946$ in geometrical approach.

$0.5878-0.5946 \,=\, -0.0068$

$\implies$ $0.5878-0.5946 \,\approx\, 0$

The difference between the values of sine of angle pi by five is $-0.0068$, which is approximately zero. However, we cannot consider the value of sine thirty six degrees, obtained from the geometrical method as its exact value because it is calculated by measuring the lengths of opposite side and hypotenuse approximately.

However, the $\sin{\Big(\dfrac{\pi}{5}\Big)}$ value is derived from a trigonometric identity in trigonometric method. Hence, we should take the value of sine of forty grades, obtained from trigonometric approach as the the exact value for the sine of angle thirty six degrees in mathematics.