(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.

 

Answer 3

Given 2 sin18 cos18, notice that this resembles double formula for sin x:

sin (2x) = 2 sin x cos x

In this case, x = 18

Thus, 2 sin18 cos 18 = sin [2 * 18] = sin 36

Answer = sin36

Answer 4

sin(A+B) = sinAcosB + cosAsinB

if A=B then

sin2A = sinA cosA + sinAcosA

sin2A = 2sinAcosA

if A=18 then

2sin18cos18 = sin2(18) = sin36

= about 0.588

using a calculator with a trig function

or

2sin18cos18= 2(.309)(.951) = 0.588

sometimes its worth doing that check on your answer

by solving it a different way, such as using a calculator

or use the small angle aproximation for 6 degrees

sinA = A if A is measured in radians

2pi = 360 degrees

6 degrees = 1/60 of 360 degrees =2pi

6 = (1/60)2pi= pi/30 =.10471

36=6(.1074)

=..644 is close to .588

both =.6 rounded off

Answer 5

simple sin(2A) = 2sinA.cosA

so 2 sin(18).cos(18)= sin (36)