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Find y such that cos y=cos(90+y) and −90degrees <= y <= 0degrees?

  1. Oct 27, 2009 #1
    1. The problem statement, all variables and given/known data
    Find y such that cos y=cos(90+y) and −90degrees <= y <= 0degrees?


    2. Relevant equations
    cos(a+b)=sin(a)cos(b)+cos(a)sin(y)


    3. The attempt at a solution

    This is a question that came up in my linear algebra class. We just learned about dot product, but this doesn't seem like it's related. It almost seems like a straight trig question.

    Anyways, is this a trick question? Wouldn't the answer be 0 since you can't take the cosine of a negative number?
     
    Last edited: Oct 27, 2009
  2. jcsd
  3. Oct 27, 2009 #2

    Mark44

    Staff: Mentor

    This is a straight trig problem, and isn't a trick question.

    The answer is NOT zero; the cosine and sine functions are defined for all real numbers. Use your relevant equations to find the solution.
     
  4. Oct 27, 2009 #3
    Okay, so plugged in:

    cos y = sin(90)cos(y)+cos(90)sin(y)

    cos(90) = 0 so that leaves

    cos y = sin(90)cos(y)

    cos(y)/cos(y) = sin(90)
    1 = sin(90), but that's already given...

    I don't know what I'm doing... don't know how to do this if there's an unknown. What do I do from here :eek:?
    Is it just simple algebra?
     
  5. Oct 27, 2009 #4
    I'm stupid, sorry. I used the wrong identity ><

    I got it
     
  6. Oct 27, 2009 #5

    Mark44

    Staff: Mentor

    So what did you get for the solution?
     
  7. Oct 28, 2009 #6
    cos(a+b) = cos(a)cos(b)-sin(a)sin(b)

    So cos(90+y) = cos(90)cos(y)-sin(90)sin(y)

    cos(y) = (0)cos(y) - (1)sin(y)

    cos(y) = -sin(y)

    1 = -sin(y)/cos(y)

    sin(y)/cos(y) = -1

    sin(y)/cos(y) = tan(y) = -1

    tan(-45) = -1, so -45

    I had to use a calculator to get the last part though. I guess I need to memorize my trig and unit circle at this point of linear algebra.
     
  8. Oct 28, 2009 #7

    Mark44

    Staff: Mentor

    Yes, you should memorize the sine and cosine of 0, 30, 45, 60, and 90 degrees (0, pi/6, pi/4, pi/3, pi/2).

    The next-to-last line you have is tan(y) = -1, so y = tan-1(-1) = -45 degrees. When you start with an equation with an unknown to find (y in this case), your last line should have the unknown in it; i.e., y = -45 degrees. Units are good, too.
     
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