SOLUTION: If cos(4°) = m, determine the value of: cos^2(178°)-cos^2(278°) in terms of m.

Algebra ->  Trigonometry-basics -> SOLUTION: If cos(4°) = m, determine the value of: cos^2(178°)-cos^2(278°) in terms of m.      Log On


   

Question 1028540: If cos(4°) = m, determine the value of:
cos^2(178°)-cos^2(278°) in terms of m.
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!

cos%5E2%28%22178%B0%22%29-cos%5E2%28%22278%B0%22%29%22%22=%22%22

cos%5E2%28%22180%B0%22-%222%B0%22%29-cos%5E2%28%22270%B0%22%2B%228%B0%22%29%22%22=%22%22

     Use the formulas:

     cos%28A-B%29=cos%28A%29cos%28B%29%2Bsin%28A%29sin%28B%29
     cos%28A%2BB%29=cos%28A%29cos%28B%29-sin%28A%29sin%28B%29




cos%28%22178%B0%22%29=%28-1%29cos%28%222%B0%22%29%2B%280%29sin%28%222%B0%22%29
cos%28%22278%B0%22%29=%280%29cos%28%228%B0%22%29-%28-1%29sin%28%228%B0%22%29

cos%28%22178%B0%22%29=-cos%28%222%B0%22%29
cos%28%22278%B0%22%29=sin%28%228%B0%22%29

%22%22=%22%22

ccos%5E2%28%222%B0%22%29-sin%5E2%28%228%B0%22%29+%22%22=%22%22
          
     We are given that cos%28%224%B0%22%29=m
 
%222%B0%22+=+expr%281%2F2%29%2A%224%B0%22,  %228%B0%22+=+2%2A%224%B0%22

cos%5E2%28expr%281%2F2%29%2A%224%B0%22%29-sin%5E2%282%2A%224%B0%22%29+%22%22=%22%22 

     Since we are given that cos%28%224%B0%22%29=m,
     we'll get everything to cosines.

     We use the identity sin%5E2%28x%29=1-cos%5E2%28x%29

cos%5E2%28expr%281%2F2%29%2A%224%B0%22%29-%281-cos%5E2%282%2A%224%B0%22%29%5E%22%22%29+%22%22=%22%22 

cos%5E2%28expr%281%2F2%29%2A%224%B0%22%29-1%2Bcos%5E2%282%2A%224%B0%22%29+%22%22=%22%22

     Next we use the half-angle and double-angle formulas:

     cos%28expr%281%2F2%29%5E%22%22x%29=+%22%22+%2B-+sqrt%28%281%2Bcos%28x%29%29%2F2%29
     cos%282x%29=2cos%5E2%28x%29-1 

%22%22=%22%22

%281%2Bcos%28%224%B0%22%29%29%2F2-1%2B%282cos%5E2%28%224%B0%22%29-1%29%5E2+%22%22=%22%22

     Substitute cos%28%224%B0%22%29=m

%281%2Bm%29%2F2-1%2B%282m%5E2-1%29%5E2+%22%22=%22%22

%281%2Bm%29%2F2-1%2B4m%5E4-4m%5E2%2B1+%22%22=%22%22

%281%2Bm%29%2F2%2B4m%5E4-4m%5E2+%22%22=%22%22

     Get an LCD of 2

%281%2Bm%29%2F2%2B8m%5E4%2F2-8m%5E2%2F2+%22%22=%22%22

%28%281%2Bm%29%2B8m%5E4-8m%5E2%29%2F2+%22%22=%22%22

%28%281%2Bm%29%2B8m%5E2%28m%5E2-1%29%29%2F2+%22%22=%22%22

%28%28m%2B1%29%2B8m%5E2%28m-1%29%28m%2B1%29%29%2F2+%22%22=%22%22

     Factor out (m+1)

%28%28m%2B1%29%281%2B8m%5E2%28m-1%29%29%29%2F2+%22%22=%22%22

%28%28m%2B1%29%281%2B8m%5E3-8m%5E2%29%29%2F2+%22%22=%22%22

%28%28m%2B1%29%288m%5E3-8m%5E2%2B1%29%29%2F2+%22%22=%22%22

     With a calculator we see that the cubic 
     trinomial in the last parentheses has a
     rational zero of 1/2:

1/2|8  -8  0  1
   |    4 -2 -1 
    8  -4 -2  0

%28%28m%2B1%29%28m-1%2F2%29%288m%5E2-4m%5E2-2%29%29%2F2+%22%22=%22%22

%28%28m%2B1%29%28m-1%2F2%292%284m%5E2-2m%5E2-1%29%29%2F2+%22%22=%22%22

%28%28m%2B1%292%28m-1%2F2%29%284m%5E2-2m%5E2-1%29%29%2F2+%22%22=%22%22

%28%28m%2B1%29%282m-1%29%284m%5E2-2m%5E2-1%29%29%2F2+
     
  That's as far as we can go.

Edwin