SOLUTION: Please help me with this equation. An equation is given. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate

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Question 1206408: Please help me with this equation.
An equation is given. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
1 − 2 sin(𝜃) = cos(2𝜃)
Find the solutions in the interval [0, 2𝜋).
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

1+-2+sin%28theta%29+=+cos%282theta%29
Find the solutions in the interval [0, 2pi).


1+-2+sin%28theta%29+=+cos%282theta%29....use identity cos%282theta%29=cos%5E2%28theta%29+-+sin%5E2%28theta%29
1+-2+sin%28theta%29+=+cos%5E2%28theta%29+-+sin%5E2%28theta%29
sin%5E2%28theta%29-cos%5E2%28theta%29+-2+sin%28theta%29+%2B1=+0...use identity cos%5E2%28theta%29=1-sin%5E2%28theta%29
sin%5E2%28theta%29-%281-sin%5E2%28theta%29%29+-2+sin%28theta%29+%2B1=+0
sin%5E2%28theta%29-1%2Bsin%5E2%28theta%29+-2+sin%28theta%29+%2B1=+0
2sin%5E2%28theta%29+-2+sin%28theta%29+=+0...factor
2sin%28theta%29%28sin%28theta%29+-1%29++=+0

solutions:
if 2sin%28theta%29+=+0 =>sin%28theta%29+=+0
if 2sin%28theta%29-1+=+0 =>sin%28theta%29+=+1%2F2

find theta
theta=sin%5E-1%280%29=> theta=0 (result in radians) or theta=0°
theta=sin%5E-1%281%2F2%29=>theta=+pi%2F6 (result in radians) ortheta=30°
the solutions in the interval [0, 2pi)
theta=0or theta=0°
theta=pi%2F6 ortheta=30°
theta=5pi%2F6 ortheta=150°
theta=pi ortheta=180°


Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
An equation is given. (Enter your answers as a comma-separated list. Let k be any integer.
Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
1 − 2 sin(𝜃) = cos(2𝜃)
Find the solutions in the interval [0, 2𝜋).
~~~~~~~~~~~~~~~~~~~~~~~


        The solution in the post by @MathLover1 is  INCORRECT.
        I came to bring a correct solution. 


1+-2+sin%28theta%29+=+cos%282theta%29

Find the solutions in the interval [0, 2pi).



1+-2+sin%28theta%29+=+cos%282theta%29....use identity cos%282theta%29=cos%5E2%28theta%29+-+sin%5E2%28theta%29

1+-2+sin%28theta%29+=+cos%5E2%28theta%29+-+sin%5E2%28theta%29

sin%5E2%28theta%29-cos%5E2%28theta%29+-2+sin%28theta%29+%2B1=+0...use identity cos%5E2%28theta%29=1-sin%5E2%28theta%29

sin%5E2%28theta%29-%281-sin%5E2%28theta%29%29+-2+sin%28theta%29+%2B1=+0

sin%5E2%28theta%29-1%2Bsin%5E2%28theta%29+-2+sin%28theta%29+%2B1=+0

2sin%5E2%28theta%29+-2+sin%28theta%29+=+0...factor

2sin%28theta%29%28sin%28theta%29+-1%29++=+0



If  sin%28theta%29+=+0  then  theta = 0  or  theta = pi radians.

If  sin%28theta%29+=+1  then  theta = pi%2F2.


ANSWER.  In the given interval, there are three and only three solutions, 

         x = 0,   x = pi%2F2 = 1.571 radians  and   x = pi = 3.142 radians (rounded as requested).

Solved (correctly)


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The post is written INCORRECTLY.

If you need the solution in the interval [0,2pi),
then you do not need to mention about "an arbitrary integer k".