SOLUTION: Find the exact value of sin(a+b), given that sin a=-4/5 and cos b = 12/13, with a in quadrant III and b in quadrant IV

Algebra ->  Trigonometry-basics -> SOLUTION: Find the exact value of sin(a+b), given that sin a=-4/5 and cos b = 12/13, with a in quadrant III and b in quadrant IV      Log On


   

Question 1127103: Find the exact value of sin(a+b), given that sin a=-4/5 and cos b = 12/13, with a in quadrant III and b in quadrant IV
Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
Use the formula 

    sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)    (1)


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


In addition to the given  sin(a) = -4%2F5  and  cos(b) = 12%2F13, you need to know  cos(a) and sin(b).


1.  cos(a) = sqrt%281-sin%5E2%28a%29%29 = sqrt%281+-+%28-4%2F5%29%5E2%29 = sqrt%281+-+16%2F25%29 = sqrt%28%2825-16%29%2F25%29 = sqrt%289%2F25%29 = -3%2F5 =-3%2F5.

   The sign "-" was chosen for the square root because cos(a) is negative when the angle "a" is in QIII.


2.  sin(b) = = sqrt%281-cos%5E2%28b%29%29 = sqrt%281+-+%2812%2F13%29%5E2%29 = sqrt%281+-+144%2F169%29 = sqrt%28169-144%29%2F169%29 = sqrt%2825%2F169%29 = -5%2F13.

   The sign "-" was chosen for the square root because sin(b) is negative when the angle "b" is in QIV.


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


sin(a+b) = %28-4%2F5%29%2A%2812%2F13%29+%2B+%28-3%2F5%29%2A%28-5%2F13%29 = -48%2F65+%2B+15%2F65 = %28-48%2B15%29%2F65 = -33%2F65.     ANSWER

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To see many other similar solved problems on calculating/evaluating 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".


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.