Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform. Experience the convenience of finding accurate answers to your questions from knowledgeable 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]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
