SOLUTION: 2 cos 2x = 1

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Question 604031: 2 cos 2x = 1
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
2cos(2x) = 1
Solving equations like this usually involve three stages:
  1. Use Trig properties and/or algebra to transform the given equation into one or more equations of the form:
    TrigFunction(Something) = SomeNumber
  2. Find the general solution(s) to each equation from stage 1.
  3. If the problem specifies that solutions are to be found only within a specified interval, then use the general solutions from stage 2 to find all the specific solutions that fall within the given interval.
Let's see this on your equation:
1. For your equation we can achieve the desired form by eliminating the 2 in front. Dividing each side by 2 we get:
cos%282x%29+=+1%2F2

2. Your equation now says that a cos is 1/2. You should recognize that 1/2 for a cos is a special value. So we will not need a calculator to find the angles whose cos is 1/2. We should know that
  • The special angle whose cos is 1/2 is pi%2F3.
  • Cos is positive in the first and 4th quadrants.
From the above we should know that any angle which has a reference angle of pi%2F3 and which terminates in the first or fourth quadrants will have the desired cos value of 1/2. There are an infinite number of these angles. Our general solution will express all of them.
1st quadrant: 2x+=+pi%2F3+%2B+2%2Api%2An (where "n" is any integer)
4th quadrant: 2x+=+%285pi%29%2F3 (or -pi%2F3) + 2%2Api%2An (where "n" is any integer)
NOTE: The left side is whatever the argument was for cos.
NOTE: The first terms on the right side of these equations can be any of the angles which terminate in that quadrant with a reference angle of pi%2F3. The ones I have written are the ones most people would use.
Now we solve for x by dividing both sides of both equations by 2:
1st quadrant: x+=+pi%2F6+%2B+pi%2An (where "n" is any integer)
4th quadrant: x+=+%285pi%29%2F6 (or -pi%2F6) + pi%2An (where "n" is any integer)
Note how each term on the right sides gets divided by 2. These equations are the general solution for your equation. By replacing the "n" in these equations with various integers, you get various specific angles that are solutions to your equation.

3. Your problem, as far as I can tell, does not ask for solutions within a given interval. So the general solution (above) is your solution. If there had been a specified interval, you would try various integer values for "n" in each of the equations to find all the solutions, if any, within the given interval.