Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Let's solve the problem step-by-step.
### Given:
- [tex]\(\cos (\alpha) = \frac{\sqrt{5}}{5}\)[/tex]
- [tex]\(\sin (\beta) = \frac{\sqrt{8}}{8}\)[/tex]
- [tex]\(\frac{\pi}{2} < \beta < \pi\)[/tex]
### (a) Find [tex]\(\sin (\alpha + \beta)\)[/tex]:
1. Find [tex]\(\sin (\alpha)\)[/tex]:
Since [tex]\(\alpha\)[/tex] is in Quadrant I, [tex]\(\sin (\alpha)\)[/tex] can be found using the Pythagorean identity:
[tex]\[ \sin^2 (\alpha) + \cos^2 (\alpha) = 1 \][/tex]
[tex]\[ \sin^2 (\alpha) + \left(\frac{\sqrt{5}}{5}\right)^2 = 1 \][/tex]
[tex]\[ \sin^2 (\alpha) + \frac{5}{25} = 1 \][/tex]
[tex]\[ \sin^2 (\alpha) + \frac{1}{5} = 1 \][/tex]
[tex]\[ \sin^2 (\alpha) = 1 - \frac{1}{5} = \frac{4}{5} \][/tex]
[tex]\[ \sin (\alpha) = \sqrt{\frac{4}{5}} = \frac{2\sqrt{5}}{5} \][/tex]
2. Find [tex]\(\cos (\beta)\)[/tex]:
Since [tex]\(\beta\)[/tex] is in Quadrant II, [tex]\(\cos (\beta)\)[/tex] is negative. Using the Pythagorean identity:
[tex]\[ \sin^2 (\beta) + \cos^2 (\beta) = 1 \][/tex]
[tex]\[ \left(\frac{\sqrt{8}}{8}\right)^2 + \cos^2 (\beta) = 1 \][/tex]
[tex]\[ \frac{8}{64} + \cos^2 (\beta) = 1 \][/tex]
[tex]\[ \frac{1}{8} + \cos^2 (\beta) = 1 \][/tex]
[tex]\[ \cos^2 (\beta) = 1 - \frac{1}{8} = \frac{7}{8} \][/tex]
[tex]\[ \cos (\beta) = -\sqrt{\frac{7}{8}} = -\frac{\sqrt{7}}{\sqrt{8}} = -\frac{\sqrt{7}}{2\sqrt{2}} = -\frac{\sqrt{14}}{4} \][/tex]
3. Use the angle addition formula for sine:
[tex]\[ \sin (\alpha + \beta) = \sin (\alpha) \cos (\beta) + \cos (\alpha) \sin (\beta) \][/tex]
Plugging in the values:
[tex]\[ \sin (\alpha + \beta) = \left(\frac{2\sqrt{5}}{5}\right) \left(-\frac{\sqrt{14}}{4}\right) + \left(\frac{\sqrt{5}}{5}\right) \left(\frac{\sqrt{8}}{8}\right) \][/tex]
[tex]\[ \sin (\alpha + \beta) = -\frac{2\sqrt{70}}{20} + \frac{\sqrt{40}}{40} \][/tex]
[tex]\[ \sin (\alpha + \beta) = -\frac{\sqrt{70}}{10} + \frac{\sqrt{10}}{10} \][/tex]
Combining the terms:
[tex]\[ \sin (\alpha + \beta) \approx -0.6785 \][/tex]
### (b) Find [tex]\(\cos (\alpha - \beta)\)[/tex]:
4. Use the angle subtraction formula for cosine:
[tex]\[ \cos (\alpha - \beta) = \cos (\alpha) \cos (\beta) + \sin (\alpha) \sin (\beta) \][/tex]
Plugging in the values:
[tex]\[ \cos (\alpha - \beta) = \left(\frac{\sqrt{5}}{5}\right) \left(-\frac{\sqrt{14}}{4}\right) + \left(\frac{2\sqrt{5}}{5}\right) \left(\frac{\sqrt{8}}{8}\right) \][/tex]
[tex]\[ \cos (\alpha - \beta) = -\frac{\sqrt{70}}{20} + \frac{2\sqrt{40}}{40} \][/tex]
[tex]\[ \cos (\alpha - \beta) = -\frac{\sqrt{70}}{20} + \frac{\sqrt{10}}{10} \][/tex]
Combining the terms:
[tex]\[ \cos (\alpha - \beta) \approx -0.1021 \][/tex]
### Final Answers:
- [tex]\(\sin (\alpha + \beta) \approx -0.6785\)[/tex]
- [tex]\(\cos (\alpha - \beta) \approx -0.1021\)[/tex]
### Given:
- [tex]\(\cos (\alpha) = \frac{\sqrt{5}}{5}\)[/tex]
- [tex]\(\sin (\beta) = \frac{\sqrt{8}}{8}\)[/tex]
- [tex]\(\frac{\pi}{2} < \beta < \pi\)[/tex]
### (a) Find [tex]\(\sin (\alpha + \beta)\)[/tex]:
1. Find [tex]\(\sin (\alpha)\)[/tex]:
Since [tex]\(\alpha\)[/tex] is in Quadrant I, [tex]\(\sin (\alpha)\)[/tex] can be found using the Pythagorean identity:
[tex]\[ \sin^2 (\alpha) + \cos^2 (\alpha) = 1 \][/tex]
[tex]\[ \sin^2 (\alpha) + \left(\frac{\sqrt{5}}{5}\right)^2 = 1 \][/tex]
[tex]\[ \sin^2 (\alpha) + \frac{5}{25} = 1 \][/tex]
[tex]\[ \sin^2 (\alpha) + \frac{1}{5} = 1 \][/tex]
[tex]\[ \sin^2 (\alpha) = 1 - \frac{1}{5} = \frac{4}{5} \][/tex]
[tex]\[ \sin (\alpha) = \sqrt{\frac{4}{5}} = \frac{2\sqrt{5}}{5} \][/tex]
2. Find [tex]\(\cos (\beta)\)[/tex]:
Since [tex]\(\beta\)[/tex] is in Quadrant II, [tex]\(\cos (\beta)\)[/tex] is negative. Using the Pythagorean identity:
[tex]\[ \sin^2 (\beta) + \cos^2 (\beta) = 1 \][/tex]
[tex]\[ \left(\frac{\sqrt{8}}{8}\right)^2 + \cos^2 (\beta) = 1 \][/tex]
[tex]\[ \frac{8}{64} + \cos^2 (\beta) = 1 \][/tex]
[tex]\[ \frac{1}{8} + \cos^2 (\beta) = 1 \][/tex]
[tex]\[ \cos^2 (\beta) = 1 - \frac{1}{8} = \frac{7}{8} \][/tex]
[tex]\[ \cos (\beta) = -\sqrt{\frac{7}{8}} = -\frac{\sqrt{7}}{\sqrt{8}} = -\frac{\sqrt{7}}{2\sqrt{2}} = -\frac{\sqrt{14}}{4} \][/tex]
3. Use the angle addition formula for sine:
[tex]\[ \sin (\alpha + \beta) = \sin (\alpha) \cos (\beta) + \cos (\alpha) \sin (\beta) \][/tex]
Plugging in the values:
[tex]\[ \sin (\alpha + \beta) = \left(\frac{2\sqrt{5}}{5}\right) \left(-\frac{\sqrt{14}}{4}\right) + \left(\frac{\sqrt{5}}{5}\right) \left(\frac{\sqrt{8}}{8}\right) \][/tex]
[tex]\[ \sin (\alpha + \beta) = -\frac{2\sqrt{70}}{20} + \frac{\sqrt{40}}{40} \][/tex]
[tex]\[ \sin (\alpha + \beta) = -\frac{\sqrt{70}}{10} + \frac{\sqrt{10}}{10} \][/tex]
Combining the terms:
[tex]\[ \sin (\alpha + \beta) \approx -0.6785 \][/tex]
### (b) Find [tex]\(\cos (\alpha - \beta)\)[/tex]:
4. Use the angle subtraction formula for cosine:
[tex]\[ \cos (\alpha - \beta) = \cos (\alpha) \cos (\beta) + \sin (\alpha) \sin (\beta) \][/tex]
Plugging in the values:
[tex]\[ \cos (\alpha - \beta) = \left(\frac{\sqrt{5}}{5}\right) \left(-\frac{\sqrt{14}}{4}\right) + \left(\frac{2\sqrt{5}}{5}\right) \left(\frac{\sqrt{8}}{8}\right) \][/tex]
[tex]\[ \cos (\alpha - \beta) = -\frac{\sqrt{70}}{20} + \frac{2\sqrt{40}}{40} \][/tex]
[tex]\[ \cos (\alpha - \beta) = -\frac{\sqrt{70}}{20} + \frac{\sqrt{10}}{10} \][/tex]
Combining the terms:
[tex]\[ \cos (\alpha - \beta) \approx -0.1021 \][/tex]
### Final Answers:
- [tex]\(\sin (\alpha + \beta) \approx -0.6785\)[/tex]
- [tex]\(\cos (\alpha - \beta) \approx -0.1021\)[/tex]
We appreciate your time. Please revisit us for more reliable answers to any questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
