2 Answers

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.

Robert J. 10/19/2012

Comments

Hi Robert,

The question was to simplify 2 sin18cos18 Wouldn't the simplest answer be sin36= sin (2x18)?

Why would I try to use the algorithm above?

Thanks.

- Rini C. 10/20/2012

Thanks for your response.

As a high school teacher who answered about 25,000 math problems in the internet, I can't believe that 2 sin 18 cos 18 = sin 36 is all you need to solve. Since 18 degree is a special angle, most likely you are asked to evaluate it. This problem can be found in many precalculus textbooks, such as in the "Avanced Mathematics" by Mr. Brown.

- Robert J. 10/20/2012

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