Let's find the exact value of [tex]\(\sin(u+v)\)[/tex] given [tex]\(\sin(u) = -\frac{3}{5}\)[/tex] for [tex]\( \frac{3\pi}{2} < u < 2\pi \)[/tex] and [tex]\(\cos(v) = \frac{15}{17}\)[/tex] where [tex]\( 0 < v < \frac{\pi}{2} \)[/tex].
### Step 1: Determine [tex]\(\cos(u)\)[/tex]
We know:
[tex]\[
\sin(u) = -\frac{3}{5}
\][/tex]
Using the Pythagorean identity:
[tex]\[
\cos^2(u) + \sin^2(u) = 1
\][/tex]
So:
[tex]\[
\cos^2(u) + \left( -\frac{3}{5} \right)^2 = 1
\][/tex]
This simplifies to:
[tex]\[
\cos^2(u) + \frac{9}{25} = 1
\][/tex]
Rearranging to solve for [tex]\(\cos^2(u)\)[/tex]:
[tex]\[
\cos^2(u) = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}
\][/tex]
Thus:
[tex]\[
\cos(u) = \pm \frac{4}{5}
\][/tex]
Given that [tex]\( \frac{3\pi}{2} < u < 2\pi \)[/tex], which is the fourth quadrant, [tex]\(\cos(u)\)[/tex] is positive:
[tex]\[
\cos(u) = \frac{4}{5}
\][/tex]
### Step 2: Determine [tex]\(\sin(v)\)[/tex]
We know:
[tex]\[
\cos(v) = \frac{15}{17}
\][/tex]
Using the Pythagorean identity:
[tex]\[
\sin^2(v) + \cos^2(v) = 1
\][/tex]
So:
[tex]\[
\sin^2(v) + \left( \frac{15}{17} \right)^2 = 1
\][/tex]
This simplifies to:
[tex]\[
\sin^2(v) + \frac{225}{289} = 1
\][/tex]
Rearranging to solve for [tex]\(\sin^2(v)\)[/tex]:
[tex]\[
\sin^2(v) = 1 - \frac{225}{289} = \frac{289}{289} - \frac{225}{289} = \frac{64}{289}
\][/tex]
Thus:
[tex]\[
\sin(v) = \pm \frac{8}{17}
\][/tex]
Given that [tex]\(0 < v < \frac{\pi}{2}\)[/tex], which is the first quadrant, [tex]\(\sin(v)\)[/tex] is positive:
[tex]\[
\sin(v) = \frac{8}{17}
\][/tex]
### Step 3: Use the angle addition formula for sine
The sine addition formula is:
[tex]\[
\sin(u+v) = \sin(u)\cos(v) + \cos(u)\sin(v)
\][/tex]
Substituting the known values:
[tex]\[
\sin(u) = -\frac{3}{5}, \quad \cos(u) = \frac{4}{5}, \quad \cos(v) = \frac{15}{17}, \quad \sin(v) = \frac{8}{17}
\][/tex]
So:
[tex]\[
\sin(u+v) = \left( -\frac{3}{5} \right) \left( \frac{15}{17} \right) + \left( \frac{4}{5} \right) \left( \frac{8}{17} \right)
\][/tex]
This simplifies to:
[tex]\[
\sin(u+v) = -\frac{45}{85} + \frac{32}{85} = -\frac{45}{85} + \frac{32}{85} = -\frac{13}{85}
\][/tex]
Therefore, the exact value of [tex]\(\sin(u+v)\)[/tex] is:
[tex]\[
- \frac{13}{17}
\][/tex]
