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).

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Question 1117291: If sin α = 4/5 and cos β = -5/13 for α in Quadrant I and β in Quadrant II, find sin(α - β).
Answer by ikleyn(52887) About Me  (Show Source):
You can put this solution on YOUR website!
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Use the formula

sin(A-B) = sin(A)*cos(B) - cos(A)*sin(B).     (*)


For it, in addition to the given values  sin(A) = 4%2F5 and cos(B) = -5%2F13  you need to find  cos(A)  and  sin(B).


1.  Since sin(A) = 4%2F5,  cos(A) = sqrt%281+-+sin%5E2%28A%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%29.

    Notice that cos(A) is positive in QI.



2.  Since cos(B) = -5%2F13,  sin(B) = sqrt%281+-+cos%5E2%28B%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%29.

    Notice that sin(B) is positive in QII.



3.  Now you have everything to use the formula (*). Substitute all given and found values into (*). You will get  

    sin(A-B) = sin(A)*cos(B) - cos(A)*sin(B) = %284%2F5%29%2A%28-5%2F13%29+-+%283%2F5%29%2A%2812%2F13%29 = -20%2F65+-+36%2F65 = -56%2F65.


Answer.   sin(A-B) = -56%2F65.

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To see many other similar solved problems, look into the lessons
    - Calculating trigonometric functions of angles
    - Advanced problems on calculating trigonometric functions of angles
    - Evaluating trigonometric expressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic  "Trigonometry: Solved problems".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.