Find cos(a+b) if sin(a) = -8/17 where "a" is in the fourth quadrant and tan(b) = -20/21 where "b" is in the second quadrant

Question

I just really struggle with these types of questions. Im not sure how to solve since there are  three trig identities. Please help!! 

Answer 1

Approach this one step at a time.

 

If sin (A) = -8/17, then what is cos A?  Hint: make a little sketch of the angle in standard position; you should have its vertical leg (coordinate) labeled -8 and the hypotenuse labeled 17.  If you remember the 8-15-17 Pythagorean triple, you'll immediately see that the horizontal leg should be labeled 15, and therefore cos A = 15/17.

 

[Alternatively, use sin²A + cos²A = 1 to calculate cos A.]

 

For Angle B, the sketch will have legs as follows:y=21, x = -20, and Pythagoras will lead you to the value of the hypotenuse, 29.  Therefore sin B = 21/29 & cos B = -20/29.

 

You now have all of the data needed to feed into cos(A+B) = cosA cosB - sinA sinB.

 

I sincerely hope that this is helpful; I strongly advise you to work hard to achieve sufficiently deep understanding & confidence, because this work is typical, and you should want to be competent by the time of the Final Exam, and into the future.