SOLUTION: If cosA = 8/17 and cosecB = 5/4 and both A and B are acute angles. Calculate the value of cos(A-B) without using a calculator.

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Question 1040635: If cosA = 8/17 and cosecB = 5/4 and both A and B are acute angles. Calculate the value of cos(A-B) without using a calculator.
Found 2 solutions by Fombitz, MathTherapy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
cos%28A%29=8%2F17
sin%28B%29=4%2F5
Use
cos%5E2%28C%29%2Bsin%5E2%28C%29=1
to find sin%28A%29 and cos%28B%29.
%288%2F17%29%5E2%2Bsin%5E2%28A%29=1
64%2F289%2Bsin%5E2%28A%29=1
sin%5E2%28A%29=%28289-64%29%2F289
sin%5E2%28A%29=225%2F289
sin%28A%29=15%2F17
and
cos%28B%29%5E2%2B%284%2F5%29%5E2=1
cos%28B%29%5E2%2B16%2F25=1
cos%28B%29%5E2=9%2F25
cos%28B%29=3%2F5
Then use the formula,
cos%28A-B%29=cos%28A%29cos%28B%29%2Bsin%28A%29sin%28B%29
Substitute,
cos%28A-B%29=%288%2F17%29%283%2F5%29%2B%2815%2F17%29%284%2F5%29
Complete that for the final answer.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
If cosA = 8/17 and cosecB = 5/4 and both A and B are acute angles. Calculate the value of cos(A-B) without using a calculator.
cos+%28A%29+=+8%2F17+=+A%2FH;                                              matrix%281%2C3%2C+csc+%28B%29+=+5%2F4%2C+or%2C+sin+%28B%29+=+4%2F5+=+O%2FH+=+y%2Fr%29

sin+%28A%29+=+O%2FH+=+y%2Fr+=+15%2F17 ----- 8-15-17 special right triangle       cos+%28B%29+=+A%2FH+=+x%2Fr+=+3%2F5 ------ 3-4-5 special right triangle



cos (A - B) = cos A cos B + sin A sin B

cos (A - B) = %288%2F17%29+%2A+%283%2F5%29+%2B+%2815%2F17%29+%2A+%284%2F5%29 

cos (A - B) = 24%2F85+%2B+60%2F85 ------- No cancellation was done since it's easier to obtain final answer with the current denominators

highlight_green%28cos+%28A+-+B%29+=+84%2F85%29