SOLUTION: Given Tan(A) = 5 in Quadrant III and Sin(B) = 2/3 in Quadrant II, find Sin(A-B),find Cos(A-B),and find the quadrant of A-B.

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Question 1105327: Given Tan(A) = 5 in Quadrant III and Sin(B) = 2/3 in Quadrant II, find Sin(A-B),find Cos(A-B),and find the quadrant of A-B.
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
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1.  tan(A) = 5  implies   = 25  implies   =  =  =   implies  sin(A) =  = .


    The sign is  "-"  (minus)  at the square root, since A is in QIII.


    Then  cos(A) =  =  = .


    The sign is  "-"  (minus)  at the square root, since A is in QIII.



2.  sin(B) =   implies  cos(B) =  =  =  =  = .


    The sign is  "-"  (minus)  at the square root, since B is in QII.



3.  Now 

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

             =  = 0.861626  (approximately).


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

             =  = -0.50754 (approximately).


    Since  sin(A-B) is positive, while cos(A-B) is negative,  the angle  A-B lies in QII.

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To see more solved similar problems on calculating trig functions,  look into the lessons
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    - Advanced problems on calculating trigonometric functions of angles
    - Evaluating trigonometric expressions
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