To find the exact value of [tex]\(\sin(u + v)\)[/tex] given that [tex]\(\sin(u) = -\frac{3}{5}\)[/tex] with [tex]\( \frac{3\pi}{2} < u < 2\pi \)[/tex] and [tex]\(\cos(v) = \frac{15}{17}\)[/tex] with [tex]\( 0 < v \)[/tex], follow these steps:
1. Determine [tex]\(\cos(u)\)[/tex]:
- Since [tex]\( \frac{3\pi}{2} < u < 2\pi \)[/tex], [tex]\(u\)[/tex] is in the fourth quadrant, where [tex]\(\sin\)[/tex] is negative and [tex]\(\cos\)[/tex] is positive.
- Use the Pythagorean identity: [tex]\(\sin^2(u) + \cos^2(u) = 1\)[/tex].
[tex]\[
\cos(u) = \sqrt{1 - \sin^2(u)}
\][/tex]
Plug in [tex]\(\sin(u) = -\frac{3}{5}\)[/tex]:
[tex]\[
\cos(u) = \sqrt{1 - \left(-\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{25}{25} - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}
\][/tex]
Since [tex]\(u\)[/tex] is in the fourth quadrant, [tex]\(\cos(u)\)[/tex] is positive:
[tex]\[
\cos(u) = \frac{4}{5}
\][/tex]
2. Determine [tex]\(\sin(v)\)[/tex]:
- Since [tex]\(0 < v\)[/tex] and [tex]\(\cos(v) = \frac{15}{17}\)[/tex], [tex]\(v\)[/tex] is in the first quadrant where both [tex]\(\sin\)[/tex] and [tex]\(\cos\)[/tex] are positive.
- Use the Pythagorean identity: [tex]\(\cos^2(v) + \sin^2(v) = 1\)[/tex].
[tex]\[
\sin(v) = \sqrt{1 - \cos^2(v)}
\][/tex]
Plug in [tex]\(\cos(v) = \frac{15}{17}\)[/tex]:
[tex]\[
\sin(v) = \sqrt{1 - \left(\frac{15}{17}\right)^2} = \sqrt{1 - \frac{225}{289}} = \sqrt{\frac{289}{289} - \frac{225}{289}} = \sqrt{\frac{64}{289}} = \frac{8}{17}
\][/tex]
3. Use the angle addition formula for sine:
The formula for [tex]\(\sin(u + v)\)[/tex] is:
[tex]\[
\sin(u + v) = \sin(u)\cos(v) + \cos(u)\sin(v)
\][/tex]
- Substitute 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]
- Compute [tex]\(\sin(u + v)\)[/tex]:
[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]
[tex]\[
\sin(u + v) = -\frac{45}{85} + \frac{32}{85} = -\frac{45 - 32}{85} = -\frac{13}{85}
\][/tex]
Thus, the exact value of [tex]\(\sin(u + v)\)[/tex] is [tex]\(-\frac{13}{85}\)[/tex].
