$α = \sin 36^{o}$ is a root of which of the following equation

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$16 x^{4} + 20 x^{2} - 5 = 0$

$16 x^{4} - 20 x^{2} + 5 = 0$

$16 x^{4} - 10 x^{2} + 5 = 0$

$10 x^{4} - 10 x^{2} - 5 = 0$

answer is C.

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Detailed Solution

To determine the equation for which α = sin 36° is a root, we proceed as follows:

Let α = sin 36° = x. Thus, we represent sin 36° with the variable x.
We know that x satisfies the relation: x = (10 - 2√5) / 4.
Rewriting this, we multiply through by 4: 4x = 10 - 2√5.
Square both sides: (4x)^2 = (10 - 2√5)^2, which simplifies to:
16x² = 100 - 40√5 + 20.
Combine like terms: 16x² = 10 - 2√5.
Introducing this into a polynomial form: (8x² - 5)^2 = 5.
Expand the expression: 64x⁴ - 80x² + 25 = 5.
Simplify further to get: 64x⁴ - 80x² + 20 = 0.
Divide through by 4 for simplicity: 16x⁴ - 20x² + 5 = 0.

Hence, the equation for which α = sin 36° is a root is:

`16x⁴ - 20x² + 5 = 0`

This equation clearly demonstrates that sin 36° is a valid root. It is derived from the fundamental properties of trigonometric values for a 36-degree angle and follows a systematic approach to simplifying the relationship between sin 36° and polynomial equations.

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