SOLUTION: If sin α = 4/5 and cos β = -5/13 for α in Quadrant I and β in Quadrant II, find sin(α - β) and cos(α - β)

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Question 1032545: If sin α = 4/5 and cos β = -5/13 for α in Quadrant I and β in Quadrant II, find sin(α - β) and cos(α - β)
Found 2 solutions by ikleyn, MathTherapy:
Answer by ikleyn(52847) About Me  (Show Source):
You can put this solution on YOUR website!
.
If sin α = 4/5 and cos β = -5/13 for α in Quadrant I and β in Quadrant II, find sin(α - β) and cos(α - β)
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Use the formulas 

sin%28alpha-beta%29 = sin%28alpha%29%2Acos%28beta%29+-+cos%28alpha%29%2Asin%28beta%29    (1)

   and

cos%28alpha-beta%29 = cos%28alpha%29%2Acos%28beta%29+%2B+sin%28alpha%29%2Asin%28beta%29.   (2)


Regarding these formulas, see the lesson Addition and subtraction formulas in this site.


In addition to the given  sin%28alpha%29 = 4%2F5  and  cos%28beta%29 = -5%2F13, you need to know  cos%28alpha%29 and sin%28beta%29.


1.  cos%28alpha%29 = sqrt%281-sin%5E2%28alpha%29%29 = sqrt%281+-+%284%2F5%29%5E2%29 = sqrt%281+-+16%2F25%29 = sqrt%2825-16%29%2F25%29 = sqrt%289%2F25%29 = +3%2F5 = 3%2F5.

   The sign "+" was chosen for the square root because alpha is in Q1.


2.  sin%28beta%29 = = sqrt%281-cos%5E2%28beta%29%29 = sqrt%281+-+%28-5%2F13%29%5E2%29 = sqrt%281+-+25%2F169%29 = sqrt%28169-25%29%2F169%29 = sqrt%28144%2F169%29 = +12%2F13 = 12%2F13.

   The sign "+" was chosen for the square root because beta is in Q2.


Now all you need to do is to substitute everything into formulas (1) and (2) and make the calculations.


sin%28alpha-beta%29 = %284%2F5%29%2A%28-5%2F13%29+-+%283%2F5%29%2A%2812%2F13%29 = -20%2F65+-+36%2F65 = %28-20-36%29%2F65 = -56%2F65,   and

cos%28alpha-beta%29 = %283%2F5%29%2A%28-5%2F13%29+%2B+%284%2F5%29%2A%2812%2F13%29 = -15%2F65+%2B+48%2F65 = %28-15%2B48%29%2F65 = 33%2F65.


Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!

If sin α = 4/5 and cos β = -5/13 for α in Quadrant I and β in Quadrant II, find sin(α - β) and cos(α - β)
sin(α - β): highlight_green%28-+56%2F65%29

cos(α - β): highlight_green%2833%2F65%29