Question

find value of
$\cos(36)$
​

Answer 1

$\huge{\bigstar}}$$\huge{\boxed{\red{\mathfrak{Answer}}}}$$\huge{\bigstar}}$

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Let A = 18°

So, 5A → 90°

→ 2A + 3A = 90°

→ 2A = 90° - 3A

Taking sine on both sides,

we get

sin 2A = sin ( 90 - 3A )

sin 2A = cos 3A

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We know that ,

→ 2 sin A cos A = 4 cos³ A - 3 cos A

→ 2 sin A cos A - 4 cos³ A + 3 cos A = 0

→ cos A ( 2 sin A - 4 cos² A + 3 ) = 0

Dividing both sides by cos A = cos 18 ≠ 0,

we get,

→ 2 sin A - 4 ( 1 - sin² A ) + 3 = 0

→ 4 sin² A + 2 sin A - 1 = 0 , which is a quadratic eqñ .

So,

$sin \: A = \frac{ - 2 \: \frac{ + }{} \sqrt{ - 4(4)( - 1)} }{2(4)} \\ sin \: A = \frac{ - 2 \frac{ + }{} \: 2 \sqrt{5} }{8} \\ sin \: A = \frac{ - 1 \: \: \frac{ + }{} \: \: \sqrt{5} }{4}$

Now sin 18° is positive, as 18° lies in first quadrant

So, sin 18° = sin A = $\frac{-1 \: \frac{+}{}\: \sqrt{5}}{4}$

Now,

→ cos 36° = ${cos}^{2.18°}$

→ cos 36° = 1 - 2 sin² 18°

→ cos 36° = $1 - 2 ( \frac{\sqrt{5}-1}{4})2$

→ cos 36° = $\frac{16-2(5+1-2\sqrt{5})}{16}$

→ cos 36° = $\frac{1 + 4 \sqrt{5}}{16}$

→ cos 36° = $\frac{\sqrt{5}+1}{4}$