SOLUTION: find all values of theta in the interval 0 less then or equal to theta less then or equal to 360 satisfy the equation sin2theta= sintheta

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Question 617159: find all values of theta in the interval 0 less then or equal to theta less then or equal to 360 satisfy the equation sin2theta= sintheta
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sin%282theta%29=+sin%28theta%29
Solving equations like this usually involve three stages:
  1. USe algebra and/or Trig properties to transform the equation into one or more equations of the form:
    TrigFunction(expression) = number
    This is often the most difficult part because it often requires a good knowledge of algebra and the Trig properties to see ways to make this transformation.
  2. Find the general solution for the equation(s) from stage 1. The general solution will include all of the infinite solutions to the equation.
  3. If the problem asks for specific solutions, like yours does, use the general solution equations to find the requested specific solutions.
Let' see this on your equation.

1. Transform the equation into the desired form.
If the transformation is not obvious, then look to:
  • Use properties to make all the arguments of the Trig functions the same.
  • Use properties to make all the functions the same (or at least all sin's and cos's.
We can use the sin%282theta%29+=2sin%28theta%29cos%28theta%29 to change the argument of 2theta into arguments of theta like the right side:
2sin%28theta%29cos%28theta%29+=+sin%28theta%29
From here we can use algebra. If we subtract sin%28theta%29 from each side we will:
  • have a zero on one side of the equation; and
  • have a factorable expression (with a common factor of sin%28theta%29)
From there we can finish the transformation
2sin%28theta%29cos%28theta%29+-+sin%28theta%29+=+0
sin%28theta%29%2A%282cos%28theta%29+-+1%29+=+0
From the Zero Product Property we know that one of these factors must be zero:
sin%28theta%29+=+0 or 2cos%28theta%29+-+1+=+0
The first equation is in the desired form. Solving the second equation for cos%28theta%29 we get:
sin%28theta%29+=+0 or cos%28theta%29+=+1%2F2
Both equations are now in the desired form.

2. Find the general solution.
Since a sin value of 0 and a cos value of 1/2 are both special angle values, this should be relatively easy. Let's first look at
sin%28theta%29+=+0
We should know from our special angles that sin has a value of 0 for angles of 0 and 180 degrees and all other angles that are co-terminal with either of those. In other "words":
theta = 0 + 360n
or
theta = 180 + 360n
(Note: The "n" in these equations can be replaced with any integer. Each integer used will give an x value that is a solution to the equation. The "+ 360n" is how we say "or any co-terminal angle".)

For cos%28theta%29+=+1%2F2 we should know from our special angles that the reference angle is 60 degrees. We should also know that cos is positive in the 1st and 3th quadrants. For the 1st quadrant we can just use the reference angle in the general solution equation. For the 4th quadrant we use either the negative of the reference angle or (360 - reference angle):
theta = 60 + 360n (for the 1st quadrant)
or
theta = -60 (or 300) + 360n (for the 4th quadrant)

So the full general solution is:
theta = 0 + 360n
or
theta = 180 + 360n
or
theta = 60 + 360n
or
theta = -60 + 360n

3. Find the specific solutions (if requested).
The problem asks for solutions in the interval 0+%3C=+theta+%3C=+360. So we try different integers for "n" in each equation until we are satisfied that we have found all the solutions that are in this interval.
For the equation
theta = 0 + 360n
we should find that when n = 0 we get theta+=+0 and when n = 1 we get theta+=+360. Higher n's result in theta's that are too big and negative n's result in values too low.
For the equation
theta = 180 + 360n
we should find that when n = 0 we get theta+=+180. All other n's give us theta's that are too big or too low.For the equation
theta = 60 + 360n
we should find that when n = 0 we get theta+=+60. All other n's give us theta's that are too big or too low.
For the equation
theta = -60 + 360n
we should find that when n = 1 we get theta+=+300. All other n's give us theta's that are too big or too low. (Note: if we had used the alternate equation, theta = 300 + 360n, we would still have gotten 300 when n = 0 instead.)

In summary the specific solutions to sin%282theta%29=+sin%28theta%29 that are in the interval 0+%3C=+theta+%3C=+360 are:
0, 60, 180, 300 and 360