Let's solve for [tex]\(\sin (\theta + \phi)\)[/tex] and [tex]\(\cos (\theta + \phi)\)[/tex] given that [tex]\(\sin \theta = \frac{3}{5}\)[/tex], [tex]\(\sin \phi = -\frac{8}{17}\)[/tex], [tex]\(\theta\)[/tex] is in Quadrant II, and [tex]\(\phi\)[/tex] is in Quadrant IV.
### Step 1: Find [tex]\(\cos \theta\)[/tex]
Given that [tex]\(\theta\)[/tex] is in Quadrant II, where [tex]\(\sin \theta > 0\)[/tex] and [tex]\(\cos \theta < 0\)[/tex], we use the Pythagorean identity:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
First, calculate [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\left( \frac{3}{5} \right)^2 = \frac{9}{25}
\][/tex]
Then solve for [tex]\(\cos^2 \theta\)[/tex] (remember [tex]\(\cos \theta\)[/tex] will be negative in Quadrant II):
[tex]\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}
\][/tex]
[tex]\[
\cos \theta = -\sqrt{\frac{16}{25}} = -\frac{4}{5}
\][/tex]
### Step 2: Find [tex]\(\cos \phi\)[/tex]
Given that [tex]\(\phi\)[/tex] is in Quadrant IV, where [tex]\(\sin \phi < 0\)[/tex] and [tex]\(\cos \phi > 0\)[/tex], we use the Pythagorean identity:
[tex]\[
\sin^2 \phi + \cos^2 \phi = 1
\][/tex]
First, calculate [tex]\(\sin^2 \phi\)[/tex]:
[tex]\[
\left( -\frac{8}{17} \right)^2 = \frac{64}{289}
\][/tex]
Then solve for [tex]\(\cos^2 \phi\)[/tex] (remember [tex]\(\cos \phi\)[/tex] will be positive in Quadrant IV):
[tex]\[
\cos^2 \phi = 1 - \sin^2 \phi = 1 - \frac{64}{289} = \frac{225}{289}
\][/tex]
[tex]\[
\cos \phi = \sqrt{\frac{225}{289}} = \frac{15}{17}
\][/tex]
### Step 3: Calculate [tex]\(\sin (\theta + \phi)\)[/tex]
Using the angle addition formula for sine:
[tex]\[
\sin (\theta + \phi) = \sin \theta \cos \phi + \cos \theta \sin \phi
\][/tex]
Substitute the known values:
[tex]\[
\sin (\theta + \phi) = \left( \frac{3}{5} \right) \left( \frac{15}{17} \right) + \left( -\frac{4}{5} \right) \left( -\frac{8}{17} \right)
\][/tex]
[tex]\[
\sin (\theta + \phi) = \frac{45}{85} + \frac{32}{85} = \frac{77}{85} \approx 0.9058823529411765
\][/tex]
### Step 4: Calculate [tex]\(\cos (\theta + \phi)\)[/tex]
Using the angle addition formula for cosine:
[tex]\[
\cos (\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi
\][/tex]
Substitute the known values:
[tex]\[
\cos (\theta + \phi) = \left( -\frac{4}{5} \right) \left( \frac{15}{17} \right) - \left( \frac{3}{5} \right) \left( -\frac{8}{17} \right)
\][/tex]
[tex]\[
\cos (\theta + \phi) = -\frac{60}{85} + \frac{24}{85} = -\frac{36}{85} \approx -0.42352941176470593
\][/tex]
So the final answers are:
[tex]\[
\sin (\theta+\phi) \approx 0.9058823529411765
\][/tex]
[tex]\[
\cos (\theta+\phi) \approx -0.42352941176470593
\][/tex]
