SOLUTION: Find the exact value for sin(alpha-beta) if sin(alpha) = -5/13 and cos(beta) = -8/17 with alpha in quadrant 3 and beta in quadrant 2

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Question 1008347: Find the exact value for sin(alpha-beta) if sin(alpha) = -5/13 and cos(beta) = -8/17 with alpha in quadrant 3 and beta in quadrant 2
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
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Find the exact value for sin(alpha-beta) if sin(alpha)=-5/13 and cos(beta)=-8/17 with alpha in quadrant 3 and beta in quadrant 2
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If sin%28alpha%29 = -5%2F13, then cos%28alpha%29 = -sqrt%281+-+sin%5E2%28alpha%29%29 = -sqrt%281-%28-5%2F13%29%5E2%29 = -sqrt%28%28169-25%29%2F169%29 = -sqrt%28144%2F169%29 = -12%2F13   
          
             (the sign is "minus" because alpha is in quadrant 3).

If cos%28beta%29 = -8%2F17, then sin%28beta%29 = sqrt%281+-+cos%5E2%28beta%29%29 = sqrt%281-%28-8%2F17%29%5E2%29 = sqrt%28%28289-64%29%2F289%29 = sqrt%28225%2F289%29 = 15%2F17   
          
             (the sign is "plus" because beta is in quadrant 2).

Now use the formula sin(alpha-beta) = sin(alpha)*cos(beta)-cos(alpha)*sin(beta).

             It is a fundamental formula Trigonometry. If you don't know it, you may look into 
             the lesson Addition and subtraction formulas in this site.

Now substitute the given and the found values of cos and sin into this formula. You will get 

sin%28alpha+-+beta%29 = %28-5%2F13%29.%28-8%2F17%29 - %28-12%2F13%29.15%2F17. 

At this point I will leave you. Please complete calculations yourself.