Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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]
### 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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
