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Math Help - cosine sum

  1. #1
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    cosine sum

    Hi. I'm going through my old trig book, for review. I am stumped by this problem, not being sure what approach to take.

    Find the exact value of cos(a +b) if sin a = 3/5 and sin b = 5/13, with a in quadrant II and b in quadrant IV.
    Of course, cos(a+b) = cos a cos b - (3/5)(5/13), but I'm not sure where to go from there. I guess I could use a calculator to get the arcsines, but I don't think that is how I am supposed to solve the problem.
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  2. #2
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    Re: cosine sum

    Hello, infraRed!

    There is a typo in the problem . . .


    \text{Find the exact value of }\cos(A + B)\text{ if: }\:\begin{Bmatrix}\sin A \:=\:\frac{3}{5},\text{ Quad 2} \\ \\[-4mm] \sin B \:=\:{\color{red}-}\frac{5}{13},\text{ Quad 4} \end{Bmatrix}
    . {\color{blue}\sin\theta\text{ is }negative\text{ in quadrant 4.}}

    We can figure the cosine values like this . . .

    \sin A \:=\;\frac{3}{5} \:=\:\frac{opp}{hyp}
    \text{Angle }A\text{ is in Quad 2 with: }\,opp = 3,\:hyp= 4
    \text{Pythagorus says: }\,adj \,=\,\pm4
    \text{In Quad 2, }adj = \text{-}4\quad\text{Hence, }\cos A \,=\,\text{-}\tfrac{4}{5}


    \sin B \,=\,-\frac{5}{13} \:=\:\frac{opp}{hyp}
    \text{Angle }B\text{ is in Quad 4 with: }\,opp = \text{-}5,\:hyp = 13
    \text{Pythagorus says: }\,adj \,=\,\pm12
    \text{In Quad 4, }adj = 12 \quad \text{Hence, }\cos B \,=\,\tfrac{12}{13}

    Now you can apply that Compound Angle Formula.
    Thanks from infraRed
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  3. #3
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    Re: cosine sum

    I see it now, thanks!
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