SOLUTION: Solve the equation https://gyazo.com/9c9598aa5e71b13cfbae900cfcb97636 (equation picture link) where 0≤x≤2π

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Question 1205297: Solve the equation https://gyazo.com/9c9598aa5e71b13cfbae900cfcb97636 (equation picture link) where 0≤x≤2π
Found 3 solutions by MathLover1, ikleyn, Alan3354:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

sin%5E2%28x%29-%281%2F2%29sin%28x%29-1%2F2=0....both sides multiply by 2

2sin%5E2%28x%29-sin%28x%29-1=0...replace sin%28x%29 with u
2u%5E2-u-1=0...factor
%282+u+%2B+1%29+%28u+-+1%29=0

solutions:
%282+u+%2B+1%29+=0 =>u=-1%2F2
%28u+-+1%29=0 => u=1
replace u with sin%28x%29
sin%28x%29=-1%2F2
sin%28x%29=1

x=sin%5E-1%28-1%2F2%29 =>x=-pi%2F6
x=sin%5E-1%281%29+=>x=pi%2F2

solutions in given interval 0+%3C=+x+%3C=+2pi:
x+=+pi%2F2 =>x=90°
x+=+%287pi%29%2F6 =>x=210°
x+=+%2811pi%29%2F6+=>x=330°


Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
sin%5E2%28x%29 - %281%2F2%29%2Asin%28x%29 - 1%2F2 = 0.     Solve for   0 <= x < 2pi.
~~~~~~~~~~~~~~~~~~~~~~ 

Multiply both sides of the given equation by 2.  

You will get an equivalent equation

    2sin%5E2%28x%29+-+sin%28x%29+-+1 = 0.


Factor left side

    (2sin(x)+1) * (sin(x)-1) = 0.


Case 1.  2sin(x) + 1 = 0  --->  sin(x) = -1%2F2  --->  x = 7pi%2F6%29  or  x = 11pi%2F6.


Case 2.   sin(x) - 1 = 0  --->  sin(x) = 1  --->  x = pi%2F2.


ANSWER.  The set of solutions is  pi%2F2,  7pi%2F6  and  11pi%2F6,  in ascending order.

Solved.



Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the equation https://gyazo.com/9c9598aa5e71b13cfbae900cfcb97636 (equation picture link) where 0≤x≤2π
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sin%5E2%28x%29+-+sin%28x%29%2F2+-+1%2F2+=+0
Sub x for sin(x)
x%5E2+-+x%2F2+-+1%2F2+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-0.5x%2B-0.5+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-0.5%29%5E2-4%2A1%2A-0.5=2.25.

Discriminant d=2.25 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--0.5%2B-sqrt%28+2.25+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-0.5%29%2Bsqrt%28+2.25+%29%29%2F2%5C1+=+1
x%5B2%5D+=+%28-%28-0.5%29-sqrt%28+2.25+%29%29%2F2%5C1+=+-0.5

Quadratic expression 1x%5E2%2B-0.5x%2B-0.5 can be factored:
1x%5E2%2B-0.5x%2B-0.5+=+%28x-1%29%2A%28x--0.5%29
Again, the answer is: 1, -0.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-0.5%2Ax%2B-0.5+%29

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sin(x) = 1
x = 90 degs or pi/2
----
sin(x) = -0.5
x = 210 degs or 7pi/6
x = 330 degs or 11pi/6