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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]
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