(2sin18)(cos18)

Question

Simplify by using the double angle formula.

Answer 1

recall double angle sin(2x) =2sinx cos x,

therefore, (2sin18) (cos 18) = sin 36.

 

Answer 2

Do you want to evaluate 2sin 18 cos 18? 

There are a lot of apporaches for this problem. Here I show you one approach:

Let x = 18 degrees.

sin 2x = sin 36 = cos 54 = cos 3x, using co-function property

Expand,

2sin x cos x = cos x cos 2x - sin x sin 2x

Cancel cos x and let u = sin x = sin 18,

2u = 1 - 2u^2 - 2u^2 = 0, sinc cos 2x = 1 - 2sin^2 x

Collect all terms in one side,

4u^2 + 2u - 1 = 0

Apply quadratic formula,

u = (1/8)[-2+sqrt(4+16)] = (1/4)(-1+sqrt(5))

cos 18 = sqrt[1-u^2] = (1/4)sqrt[10+2sqrt(5)]

2sin 18 cos 18 = 2(1/4)(-1+sqrt(5))(1/4)sqrt[10+2sqrt(5)] which can be simplified further.

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Attn: Since sin 18 is positive, u = (1/8)[-2-sqrt(4+16)] is not a solution.