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.

Algebra ->  Trigonometry-basics -> 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.      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
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1.  tan(A) = 5  implies  tan%5E2%28A%29 = 25  implies  sin%5E2%28A%29 = 1%2F%281+%2B+1%2Ftan%5E2%28A%29%29 = 1%2F%281+%2B+1%2F25%29 = 25%2F26  implies  sin(A) = -sqrt%2825%2F26%29 = -5%2Fsqrt%2826%29.


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


    Then  cos(A) = -sqrt%281-sin%5E2%28A%29%29 = -sqrt%281-25%2F26%29 = -1%2Fsqrt%2826%29.


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



2.  sin(B) = 2%2F3  implies  cos(B) = -sqrt%281-sin%5E2%28B%29%29 = -sqrt%281-%282%2F3%29%5E2%29 = -sqrt%281-4%2F9%29 = -sqrt%285%2F9%29 = -sqrt%285%29%2F3.


    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) = %28-5%2Fsqrt%2826%29%29%2A%28-sqrt%285%29%2F3%29 - %28-1%2Fsqrt%2826%29%29%2A%282%2F3%29 = %285%2Asqrt%285%29%29%2F%283%2Asqrt%2826%29%29 + 2%2F%283%2Asqrt%2826%29%29 = %285%2Asqrt%285%29%2B2%29%2F%283%2Asqrt%2826%29%29 = %285%2Asqrt%285%29%2B2%29%2F%283%2Asqrt%2826%29%29.sqrt%2826%29%2Fsqrt%2826%29 = %285%2Asqrt%28130%29%2B2%2Asqrt%2826%29%29%2F%283%2A26%29 = 

             = %285%2Asqrt%28130%29%2B2%2Asqrt%2826%29%29%2F78 = 0.861626  (approximately).


    cos(A-B) = cos(A)*cos(B) + sin(A)*sin(B) = %28-1%2Fsqrt%2826%29%29%2A%28-sqrt%285%29%2F3%29 + %28-5%2Fsqrt%2826%29%29%2A%282%2F3%29 = %28sqrt%285%29%29%2F%283%2Asqrt%2826%29%29 - 10%2F%283%2Asqrt%2826%29%29 = %28sqrt%285%29-10%29%2F%283%2Asqrt%2826%29%29 = %28sqrt%285%29-10%29%2F%283%2Asqrt%2826%29%29.sqrt%2826%29%2Fsqrt%2826%29 = %28sqrt%28130%29-10%2Asqrt%2826%29%29%2F%283%2A26%29 = 

             = %28sqrt%28130%29-10%2Asqrt%2826%29%29%2F78 = -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
    - 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".


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

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