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To solve for [tex]\(\sin(u+v)\)[/tex] and [tex]\(\cos(u+v)\)[/tex], given [tex]\(\sin(u) = \frac{12}{13}\)[/tex] and [tex]\(\cos(v) = -\frac{15}{17}\)[/tex] with [tex]\(\frac{\pi}{2} \leq u \leq \pi\)[/tex] and [tex]\(\frac{\pi}{2} \leq v \leq \pi\)[/tex], we will use trigonometric identities and information about the signs of sine and cosine in different quadrants.
### Step-by-Step Solution
1. Determine [tex]\(\cos(u)\)[/tex]:
Since [tex]\(u \in [\frac{\pi}{2}, \pi]\)[/tex], it's in the second quadrant where [tex]\(\sin(u)\)[/tex] is positive and [tex]\(\cos(u)\)[/tex] is negative.
[tex]\[ \cos(u) = -\sqrt{1 - \sin^2(u)} = -\sqrt{1 - \left( \frac{12}{13} \right)^2} \][/tex]
[tex]\[ \cos(u) = -\sqrt{1 - \frac{144}{169}} = -\sqrt{\frac{25}{169}} = -\frac{5}{13} \][/tex]
2. Determine [tex]\(\sin(v)\)[/tex]:
Since [tex]\(v \in [\frac{\pi}{2}, \pi]\)[/tex], it's in the second quadrant where [tex]\(\cos(v)\)[/tex] is negative and [tex]\(\sin(v)\)[/tex] is positive.
[tex]\[ \sin(v) = \sqrt{1 - \cos^2(v)} = \sqrt{1 - \left( -\frac{15}{17} \right)^2} \][/tex]
[tex]\[ \sin(v) = \sqrt{1 - \frac{225}{289}} = \sqrt{\frac{64}{289}} = \frac{8}{17} \][/tex]
3. Compute [tex]\(\sin(u + v)\)[/tex] using the sine addition formula:
[tex]\[ \sin(u + v) = \sin(u)\cos(v) + \cos(u)\sin(v) \][/tex]
Substitute the values we have:
[tex]\[ \sin(u + v) = \left( \frac{12}{13} \right)\left( -\frac{15}{17} \right) + \left( -\frac{5}{13} \right)\left( \frac{8}{17} \right) \][/tex]
[tex]\[ \sin(u + v) = -\frac{180}{221} - \frac{40}{221} = -\frac{220}{221} \][/tex]
[tex]\[ \sin(u + v) = -0.9954751131221719 \][/tex]
4. Compute [tex]\(\cos(u + v)\)[/tex] using the cosine addition formula:
[tex]\[ \cos(u + v) = \cos(u)\cos(v) - \sin(u)\sin(v) \][/tex]
Substitute the values we have:
[tex]\[ \cos(u + v) = \left( -\frac{5}{13} \right)\left( -\frac{15}{17} \right) - \left( \frac{12}{13} \right)\left( \frac{8}{17} \right) \][/tex]
[tex]\[ \cos(u + v) = \frac{75}{221} - \frac{96}{221} = -\frac{21}{221} \][/tex]
[tex]\[ \cos(u + v) = -0.09502262443438936 \][/tex]
### Results
- a) [tex]\(\sin(u + v) = -0.9954751131221719\)[/tex]
- b) [tex]\(\cos(u + v) = -0.09502262443438936\)[/tex]
These values are the results for [tex]\(\sin(u + v)\)[/tex] and [tex]\(\cos(u + v)\)[/tex] respectively.
### Step-by-Step Solution
1. Determine [tex]\(\cos(u)\)[/tex]:
Since [tex]\(u \in [\frac{\pi}{2}, \pi]\)[/tex], it's in the second quadrant where [tex]\(\sin(u)\)[/tex] is positive and [tex]\(\cos(u)\)[/tex] is negative.
[tex]\[ \cos(u) = -\sqrt{1 - \sin^2(u)} = -\sqrt{1 - \left( \frac{12}{13} \right)^2} \][/tex]
[tex]\[ \cos(u) = -\sqrt{1 - \frac{144}{169}} = -\sqrt{\frac{25}{169}} = -\frac{5}{13} \][/tex]
2. Determine [tex]\(\sin(v)\)[/tex]:
Since [tex]\(v \in [\frac{\pi}{2}, \pi]\)[/tex], it's in the second quadrant where [tex]\(\cos(v)\)[/tex] is negative and [tex]\(\sin(v)\)[/tex] is positive.
[tex]\[ \sin(v) = \sqrt{1 - \cos^2(v)} = \sqrt{1 - \left( -\frac{15}{17} \right)^2} \][/tex]
[tex]\[ \sin(v) = \sqrt{1 - \frac{225}{289}} = \sqrt{\frac{64}{289}} = \frac{8}{17} \][/tex]
3. Compute [tex]\(\sin(u + v)\)[/tex] using the sine addition formula:
[tex]\[ \sin(u + v) = \sin(u)\cos(v) + \cos(u)\sin(v) \][/tex]
Substitute the values we have:
[tex]\[ \sin(u + v) = \left( \frac{12}{13} \right)\left( -\frac{15}{17} \right) + \left( -\frac{5}{13} \right)\left( \frac{8}{17} \right) \][/tex]
[tex]\[ \sin(u + v) = -\frac{180}{221} - \frac{40}{221} = -\frac{220}{221} \][/tex]
[tex]\[ \sin(u + v) = -0.9954751131221719 \][/tex]
4. Compute [tex]\(\cos(u + v)\)[/tex] using the cosine addition formula:
[tex]\[ \cos(u + v) = \cos(u)\cos(v) - \sin(u)\sin(v) \][/tex]
Substitute the values we have:
[tex]\[ \cos(u + v) = \left( -\frac{5}{13} \right)\left( -\frac{15}{17} \right) - \left( \frac{12}{13} \right)\left( \frac{8}{17} \right) \][/tex]
[tex]\[ \cos(u + v) = \frac{75}{221} - \frac{96}{221} = -\frac{21}{221} \][/tex]
[tex]\[ \cos(u + v) = -0.09502262443438936 \][/tex]
### Results
- a) [tex]\(\sin(u + v) = -0.9954751131221719\)[/tex]
- b) [tex]\(\cos(u + v) = -0.09502262443438936\)[/tex]
These values are the results for [tex]\(\sin(u + v)\)[/tex] and [tex]\(\cos(u + v)\)[/tex] respectively.
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