Respuesta :
We are given the following information
sin u = 5/13
cos v = -3/5
Where the angle u and v are in the 2nd quadrant.
[tex]\begin{gathered} \cos\theta=\frac{adjacent}{hypotenuse} \\ \sin\theta=\frac{opposite}{hypotenuse} \end{gathered}[/tex]Let us find cos u
Apply the Pythagorean theorem to find the 3rd side.
[tex]\begin{gathered} a^2+b^2=c^2 \\ a^2=c^2-b^2 \\ a^2=13^2-5^2 \\ a^2=169-25 \\ a^2=144 \\ a=\sqrt{144} \\ a=12 \end{gathered}[/tex]Cos u = 12/13
Now, let us find sin v
Apply the Pythagorean theorem to find the 3rd side.
[tex]\begin{gathered} a^2+b^2=c^2 \\ b^2=c^2-a^2 \\ b^2=5^2-(-3)^2 \\ b^2=25-9 \\ b^2=16 \\ b=\sqrt{16} \\ b=4 \end{gathered}[/tex]Sin v = 4/5
Recall the formula for sin (A - B)
[tex]\sin(A-B)=\sin A\cos B-\cos A\sin B[/tex]Let us apply the above formula to the given expression
[tex]\begin{gathered} \sin(u-v)=\sin u\cdot\cos v+\cos u\cdot\sin v \\ \sin(u-v)=\frac{5}{13}\cdot-\frac{3}{5}+\frac{12}{13}\cdot\frac{4}{5} \\ \sin(u-v)=\frac{33}{65} \end{gathered}[/tex]Therefore, sin (u - v) = 33/65
