$\sin 18^\circ = \dfrac{\sqrt{5} -1}{4} = 0.309016994\ldots$

Theoretically, the value of sine of angle $18^\circ$ can be derived in mathematics by the product rule and some trigonometric formulas.

We know that $5 \times 18$ is equal to $90$. Therefore $5 \times 18^\circ$ is equal to $90^\circ$. On the basis of this rule, the value of $\sin 18^\circ$ can be derived in mathematics exactly.

Assume $\theta = 18^\circ$. Therefore $5 \theta = 90^\circ$.

Split the angle $5 \theta$ as a compound angle. It is better to write it as the sum of $2 \theta$ and $3 \theta$.

$2 \theta + 3 \theta = 90^\circ$

Move $3 \theta$ to right hand side of the equation.

$2 \theta = 90^\circ -3 \theta $

Now take sine both sides.

$\sin (2 \theta) = \sin(90^\circ -3 \theta)$

$\implies \sin 2\theta = \cos 3\theta$

Expand sine of double angle and cosine of triple angle in terms of the angle theta.

$\implies 2\sin \theta \cos \theta = 4\cos^3 \theta -3\cos \theta$

$\implies 0 = 4\cos^3 \theta -3\cos \theta -2\sin \theta \cos \theta$

$\implies 4\cos^3 \theta -3\cos \theta -2\sin \theta \cos \theta = 0$

$cos \theta$ is a common multiplying factor in each term of this equation. So, take common from each term.

$\implies \cos \theta (4\cos^2 \theta -3 -2\sin \theta) = 0$

According to Pythagorean trigonometric identity rule, the square of cosine can be transformed in terms of sine. It helps the equation to become a quadratic equation.

$\implies \cos \theta (4(1 -\sin^2 \theta) -3 -2\sin \theta) = 0$

$\implies \cos \theta (4 -4\sin^2 \theta -3 -2\sin \theta) = 0$

$\implies \cos \theta (-4\sin^2 \theta -2\sin \theta + 1) = 0$

$\implies \cos \theta (4\sin^2 \theta + 2\sin \theta -1) = 0$

The cubic function in terms of trigonometric ratios is now transformed as two multiplying factors and the product of them is equal to zero.

Therefore, $\cos \theta = 0$ and $4\sin^2 \theta + 2\sin \theta -1 = 0$

In this case, the angle theta is assumed $18$ degrees. The value of cosine is zero at angle $90^\circ$. It means, $\cos 90^\circ = 0$. So, it is not possible $\cos 18^\circ$ is equal to $0$. Hence, $\cos 18^\circ \ne 0$ and ignore $\cos \theta = 0$ equation.

Now consider the second multiplying factor ($4\sin^2 \theta + 2\sin \theta -1 = 0$), which is a quadratic equation in terms of sine. Use quadratic equation formula method and obtain both roots of this equation.

$\sin \theta = \dfrac{-2 \pm \sqrt{(2)^2 -4 \times 4 \times (-1)}}{2 \times 4}$

$\implies \sin \theta = \dfrac{-2 \pm \sqrt{4 + 16}}{8}$

$\implies \sin \theta = \dfrac{-2 \pm \sqrt{20}}{8}$

$\implies \sin \theta = \dfrac{-2 \pm 2\sqrt{5}}{8}$

$\implies \sin \theta = \dfrac{-1 \pm \sqrt{5}}{4}$

$\sin \theta = \dfrac{-1 -\sqrt{5}}{4}$ and $\sin \theta = \dfrac{-1 + \sqrt{5}}{4}$ are two values of sine but one of them is true and other is not.

$\sin \theta = \dfrac{-1 -\sqrt{5}}{4} = -0.809016994\ldots$

Geometrically, the length of opposite side is less than or equal to the length of the hypotenuse in a right angled triangle. So, the ratio of length of the opposite side to length of hypotenuse is less than or equal to one. The value of $\sin 18^\circ$ can be $-0.809016994\ldots$

The right angled triangle lies in first quadrant when the angle of the triangle is $18^\circ$. In first quadrant, the $x$-axis and $y$-axis represent positive values.

Therefore, the length of the opposite side and length of hypotenuse are positive. Hence therefore the value of $\sin 18^\circ$ is a positive value.

$\therefore \,\,\,\, \sin \theta \ne \dfrac{-\sqrt{5} -1}{4}$

Now, consider the second value of sine function.

$\sin \theta = \dfrac{\sqrt{5} -1}{4}$

$\implies \sin 18^\circ = \dfrac{\sqrt{5} -1}{4} = 0.309016994\ldots$

In this case, the value of $\sin 18^\circ$ is positive and less than one. So, the value of $\sin 18^\circ$ is $\dfrac{\sqrt{5} -1}{4}$.

The value of sine $18$ degrees can be verified geometrically.

- Draw a horizontal line of any length. For example, $9$ cm length horizontal line is drawn by the ruler and it is called $\overline{EF}$.
- Draw a perpendicular line at point $F$ by using protractor.
- Use protractor and draw a line with $18$ degrees angle at $E$ of the horizontal line.
- The perpendicular line and $18^\circ$ angle line both are intersected at a point which is called as point $G$ and thus the right angled triangle $FEG$ is constructed geometrically.
- Now measure lengths of opposite side and hypotenuse by the ruler and they are roughly measured as $2.95$ cm and $9.45$ cm respectively.

Calculate the ratio of length of the opposite side to length of the hypotenuse to find the value of $\sin 18^\circ$ in geometric approach.

$\sin 18^\circ = \dfrac{FG}{EG}$

$\implies \sin 18^\circ = \dfrac{2.95}{9.45} = 0.312169312\ldots$

The value of sine $18$ degrees is $0.309016994\ldots$ in theoretical approach but it is $0.312169312\ldots$ in geometrical approach. However, the both values are approximately same.

Unfortunately, the lengths of opposite side and hypotenuse in $\Delta FEG$ are not exact values. So, the chance of obtaining the exact value of $\sin 18^\circ$ in geometric method is very less. Due to this, the value of sine $18$ degrees in geometric method cannot be considered as exact value.

However, the sine $18$ degrees value, obtained from theoretical method is exact value. It is actually derived purely in mathematical and trigonometrical methods, and there is no chance for error.

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