Expert: Paul Klarreich - 9/13/2009

Question
I am really having trouble understanding how to find exact solutions over intervals?  The one I am currently stuck on is:

cos 2x=-1/2 over the intervals (0, 2pi).

This is as far as I have gotten:

0<x<2pi = 0<2x<4pi.

Thanks a bunch for the help.

Answer
Questioner: Patricia
Country: United States
Category: Advanced Math
Private: No
Subject: Exact Solutions
Question: I am really having trouble understanding how to find exact solutions over intervals?  The one I am currently stuck on is:

cos 2x=-1/2 over the intervals (0, 2pi).

This is as far as I have gotten:

0<x<2pi = 0<2x<4pi.

>>> You should say:  If:  0<x<2pi,  then:  0<2x<4pi.

Thanks a bunch for the help.
........................................
Yes, that is correct, but to go further, you should:

1. Learn about the 'special' angles:

30,45,60 degrees, or
pi/6, pi/4, pi/3  radians.

And that cos 60 = 1/2.

2. Handle angles in the four quadrants, so you would know that:

cos t = (neg) when  t is in Q2,Q3.

Then you will say:

cos t = -1/2 when  t[ref] = 60, and in

Q2:  t = 120 deg = 2pi/3
Q3:  t = 240 deg = 4pi/3

And, since you DID say that  0 < t < 4pi, you can add 2pi to each of those values:

Q2:  t = 120 deg = 2pi/3  or  8pi/3
Q3:  t = 240 deg = 4pi/3  or  10pi/3

But t = 2x, so you can now write:

2x = 120 deg = 2pi/3  or  8pi/3
2x = 240 deg = 4pi/3  or  10pi/3

and solve for x:

x = 60 deg = pi/3  or  4pi/3
x = 120 deg = 2pi/3  or  5pi/3