Question

angles A and B are located in the first quadrant. if sin A= 5/13 and cos B = 3/5,determine the exact value of cos (A+B)

Answer 1

well draw the 2 triangles...



find x and y using pythagoras' theorem

then you need to know

\[\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)\]

so make the ratio substitutions and then calculate an answer

Answer 2

use the pythagorean theorem to find "x" and "y" for each

once you find the missing side for each
you'd find the cosine and sine for each corresponding

Answer 3

for example, notice campbell_st picture
thus we could say \(\bf sin(\theta)=\cfrac{opposite}{hypotenuse}=\cfrac{b}{c}
\qquad \qquad
% cosine
cos(\theta)=\cfrac{adjacent}{hypotenuse}=\cfrac{a}{c}
\\ \quad \\
sin(A)=\cfrac{5}{13}\to \cfrac{b}{c}\to \cfrac{b=5}{c=13}\impliedby \textit{find cos(A) first}
\\ \quad \\
c^2=a^2+b^2\implies c^2-b^2=a^2\implies \pm\sqrt{c^2-b^2}={\color{blue}{ a}}
\\ \quad \\
thus\implies cos(A)=\cfrac{{\color{blue}{ a}}}{c}\)


the one thing about the pythagorean theorem is that
the root could give us either, positive or negative values
so, which one do we use?

well, we know that angle A is in the 1st quadrant
and in the 1st quadrant, side "x" or the adjacent side, is positive
so we use the positive value from the pythagorean theorem

Answer 4

and then you do the same for the missing side on angle B
to get the sin(B) for it

once you get those two, cos(A) and sin(B)
use as campbell_st suggested -> \(\bf cos({\color{brown}{ \alpha}} + {\color{blue}{ \beta}})= cos({\color{brown}{ \alpha}})cos({\color{blue}{ \beta}})- sin({\color{brown}{ \alpha}})sin({\color{blue}{ \beta}})\)