Proofs of $\sin{(15^\circ)}$ value
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The value of sine of angle fifteen degrees can be derived mathematically in two different methods possibly. So, let us learn the mathematical approaches to know how to derive the value of sine of fifteen degrees in mathematics.
Trigonometric Proof
$\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{6}-\sqrt{2}}{4}$
The exact value of sine of angle fifteen degrees is square root of six minus square root of two divided by four. The sine of $15^\circ$ value in fraction form can be derived exactly in mathematical form by using the trigonometric identities.
Geometric Proof
$\sin{(15^\circ)} \,=\, 0.253$
The value of sine of $15^\circ$ can also be derived mathematically in geometric system by constructing a right triangle with angle of fifteen degrees but the value of sine of fifteen degrees can only be derived in decimal form.
Conclusion
$(1).\,\,\,$ $\sin{(15^\circ)}$ $\,=\,$ $\dfrac{\sqrt{3}-1}{2\sqrt{2}}$ $\,=\,$ $0.2588$
$(2).\,\,\,$ $\sin{(15^\circ)}$ $\,=\,$ $0.253$
According to the trigonometric method, it is derived that the exact value of sine of angle $15$ degrees in fraction form is square root of three minus one divided by two times square root of two. Therefore, the value of sine of $15$ degrees in decimal form is approximately equal to $0.2588$. However, it is derived that the value of sine of angle fifteen degrees is approximately equal to $0.253$ as per the geometric method.
Due to the technical difficulties in measuring the lengths of opposite side and hypotenuse, the value of sine of fifteen degrees, which is obtained in geometric system is approximately equal to the actual value, obtained from trigonometric system. Therefore, the value of sine $15$ degrees is $0.253$, which cannot be considered as actual value.
