
Shawn L. answered 10/18/12
Master math tutor: excellent problem solver
recall double angle sin(2x) =2sinx cos x,
therefore, (2sin18) (cos 18) = sin 36.
Erin K.
asked 10/18/12Simplify by using the double angle formula.
Shawn L. answered 10/18/12
Master math tutor: excellent problem solver
recall double angle sin(2x) =2sinx cos x,
therefore, (2sin18) (cos 18) = sin 36.
Robert J. answered 10/19/12
Certified High School AP Calculus and Physics Teacher
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.
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.
10/20/12
Arturo O.
sin(2x) = 2 sin(x) cos(x) is an identity, which is true for all x.12/08/19
sin(A+B) = sinAcosB + cosAsinB
if A=B then
sin2A = sinA cosA + sinAcosA
sin2A = 2sinAcosA
if A=18 then
2sin18cos18 = sin2(18) = sin36
Sumit K. answered 07/30/20
simple sin(2A) = 2sinA.cosA
so 2 sin(18).cos(18)= sin (36)
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Rini C.
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.
10/20/12