SOLUTION: if a+b=90degree, then the maximum value of cosacosb is..

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Question 305319: if a+b=90degree, then the maximum value of cosacosb is..
Found 2 solutions by toidayma, Edwin McCravy:
Answer by toidayma(44) About Me  (Show Source):
You can put this solution on YOUR website!
For any a,b we have:
cos(a) * cos(b) = %281%2F2%29%2A%28cos%28a%2Bb%29+%2B+cos%28a-b%29%29
Since a + b = Pi/2, therefore
cos(a) * cos(b) = %281%2F2%29%2A%28cos%28Pi%2F2%29+%2B+cos%28a-b%29%29+=+%281%2F2%29%2Acos%28a-b%29 since cos(Pi/2) = 0.
Since cos%28a-b%29+%3C=+cos%280%29+=+1, "=" occurs when a = b = Pi/4 = 45 degrees.
Thus, the maximum value of cos(a)*cos(b) is 1.

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
if a+b=90degree, then the maximum value of cosacosb is..

Let a = x, since the letters x is usually a variable, and 
letters a and b are usually constants.  Then b = 90°-a = 90°-x 

I will approach it from a calculus standpoint.


y = cos(x)cos(90-x)

y = cos(x)sin(x)



%28dy%29%2F%28dx%29+=+Cos%282x%29

Setting that = 0

Cos(2x)=0

2x = 90°, 270°, 450°, etc.

x = 45°, 135°, 225°, etc.

Substituting these in 

y = cos(x)sin(x)

y = cos(45°)sin(45°) = %28sqrt%282%29%2F2%29%2A%28sqrt%282%29%2F2%29=2%2F4+=+1%2F2

y = cos(135°)sin(135°) = %28-sqrt%282%29%2F2%29%2A%28sqrt%282%29%2F2%29=-2%2F4+=+-1%2F2

y = cos(225°)sin(225°) = %28-sqrt%282%29%2F2%29%2A%28-sqrt%282%29%2F2%29=2%2F4+=+1%2F2

etc.

So the maximum value is 1%2F2 and the minimum value is -1%2F2.

Edwin