SOLUTION: I have tried so many times to solve the problem Find the exact value of sin (x/2) = 2cos^2(x)-1 I've changed the sin (x/2) to the half angle identity and changed the 2cos^2(

Question 1002887: I have tried so many times to solve the problem
Find the exact value of
sin (x/2) = 2cos^2(x)-1
I've changed the sin (x/2) to the half angle identity and changed the 2cos^2(x)-1 to cos 2x, which is the double angle identity and tried to solve to be able to set everything to 0, but nothing I have tried worked. Please help as best as you can.

Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Find the exact value of
sin (x/2) = 2cos^2(x)-1
------------
The trig functions have to have the same argument.
sin^2(x/2) = (1 - cos(x))/2
----
sin(x/2) = 2cos^2(x)-1
Square both sides
sin^2(x/2) = 4cos^4(x) - 4cos^2(x) + 1
(1 - cos(x))/2 = 4cos^4(x) - 4cos^2(x) + 1
1 - cos(x) = 8cos^4(x) - 8cos^2(x) + 2
8cos^4(x) - 8cos^2(x) + cos(x) + 1 = 0
----------------
If you inspect
8x^4 - 8x^2 + x + 1 = 0 it's obvious that x = -1 is a zero.
--> cos(x) = -1
--> x = pi + n*2pi, n = �0,1,2,3...
------------
Divide the quartic by (x+1)
--> 8x^3 - 8x^2 + 1 = 0
It's less obvious, x = 1/2 is a zero.
--> cos(x) = 0.5
--> x = pi/3 + n*2pi, n = �0,1,2,3...
and
--> x = 2pi/3 + n*2pi, n = �0,1,2,3...
-------------------------
Divide the cubic by (x - 1/2):
--> 8x^2 - 4x - 2 = 0

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=80 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 0.809016994374947, -0.309016994374947. Here's your graph:

======================
--> cos(x) =
--> x = pi/5 + n*2pi, n = �0,1,2,3...
--> x = 9pi/5 + n*2pi, n = �0,1,2,3...
-----------------
and x = 3pi/5 + n*2pi, n = �0,1,2,3...
and x = 7pi/5 + n*2pi, n = �0,1,2,3...

RELATED QUESTIONS
if f(x)=2cos^2 x+sin x-1 find the value of f(pi/2) (answered by stanbon)

please help me solve this equation: solve exactly: sin(x) + 4csc(x) + 5=0 I have... (answered by stanbon)

Solve the equation for x, on the interval 0 is less than or equal to x and 2pi is greater (answered by Alan3354)

Solve the equation sin^2x (plus) 2cos^2x=2.find the value of... (answered by mananth)

sin(x)/2cos(X)/2(1-2sin^2(x)/2) The answer in my book is 1/4 sin(2x) I tried solving... (answered by Alan3354,ikleyn)

Prove: 1-cot^2x/1+cot^2x=2cos^2-1 I tried this: LHS... (answered by Alan3354)

how to verify the identity of problem... (answered by lwsshak3)

Give sin(a) =8/9, pi/2< a < pi, Find the exact value of sin(a/2) I tried solving it... (answered by jim_thompson5910)

Hi! Given that x^2-14x+a=(x+b)^2 for all values of x, find the value of a and the... (answered by reviewermath,Okeke_Christian)