Math Help - cosine sum

1. 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.

2. 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.

3. Re: cosine sum

I see it now, thanks!