Question

what is the value of Cos 36° ?

In detail.......​

Answer 1

Answer:

If you calculate it aiding 991ES calculator then the answer comes to

0.809016...(approx)

Answer 2

To find:-

cos36° = ?

Solution:-

Let β = 18° .Then,

β = 18° => 5β = 90°

⠀⠀⠀ ⠀=> 3β+2β = 90°

⠀⠀⠀ ⠀=> 2β = 90° - 3β

⠀⠀ ⠀⠀=> sin2β = sin(90°-3β) = cos3β

⠀⠀⠀⠀ => 2sinβ cosβ = 4cos³β - 3cosβ

⠀ ⠀=> 2sinβcosβ - 4cos³β + 3cosβ = 0

⠀⠀⠀⠀ => cosβ(2sinβ - 4cos²β + 3) = 0

⠀⠀ ⠀⠀=> 2sinβ - 4cos²β + 3 = 0

⠀⠀⠀⠀ => 2sinβ - 4(1-sin²β) + 3 = 0

⠀⠀ ⠀=> 4sin²β + 2sinβ -1 = 0

According to Sri Dharacharya formula

⠀⠀⠀⠀

$= > sin \beta = \frac{ - 2 ± \sqrt{4 + 16} }{8} \\ = > sin \beta = \frac{ - 2 ± \sqrt{20} }{8} = \frac{ - 2 ± 2 \sqrt{5} }{8} \\ = > sin \beta = \frac{2( -1 ± \sqrt{5}) }{8} = > \frac{ \sqrt{5} - 1 }{4} \\ = > sin \beta = \frac{ \sqrt{5} - 1 }{4}$

As we are supposed that β= 18°

Then,

$sin18 = \frac{ (\sqrt{5} - 1) }{4}$

Now for cos18°

=> cos36° = cos(2×18) = (1 - 2sin²18°)

=> cos36° = $1 - 2 \frac{ {( \sqrt{5 } - 1)}^{2} }{16} = 1 - \frac{6 - 2 \sqrt{5} }{8}$

$= >cos36 = \frac{ \sqrt{5} + 1}{4}$

Hence proved.❤

Hope its help uh❤