Animated Solution for Mathematics - Trigonometry: $\alpha =\sin 36^{\circ }$ is a root of which of the following equation

Welcome, future engineers! Today, we are embarking on a journey to uncover the hidden algebraic identity of $\sin 36^{\circ }$.

In the world of JEE Advanced, numbers are stories waiting to be told. We start by defining our variable as $\alpha =\sin 36^{\circ }$.

Our mission is to find a polynomial $f(x)$ such that $f(\alpha )=0$. The secret lies in the geometry of the circle, specifically noting that $72^{\circ }$ is the bridge because it is $2\times 36^{\circ }$.

Double Angle Identity

Using the identity $\cos 2\theta =1-2{\sin }^2\theta$, we can express $\cos 72^{\circ }$ in terms of our variable:

Now, we invoke the standard trigonometric value for this angle:

Equating these two expressions, we obtain the following relationship:

Now, the algebra begins. Multiply the entire equation by 4 to clear the denominator:

Rearranging the terms to isolate the radical gives:

This is the moment of truth. To banish the radical, we square both sides of the equation:

Expanding the left side, we get:

Subtracting 5 from both sides yields the equation:

Dividing the entire equation by 4, we arrive at the final polynomial:

This is the polynomial! It is elegant, precise, and powerful. You have just derived a fundamental property of the golden ratio's cousin. Keep this spirit of inquiry alive!