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Sagot :
Certainly! Let's tackle each part of the question step by step.
### Part (a)
Given: [tex]\(\sin \theta = \frac{8}{17}\)[/tex]
1. Calculating [tex]\(\cos \theta\)[/tex]:
We use the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substituting [tex]\(\sin \theta = \frac{8}{17}\)[/tex]:
[tex]\[ \left(\frac{8}{17}\right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{64}{289} + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{64}{289} \][/tex]
[tex]\[ \cos^2 \theta = \frac{289}{289} - \frac{64}{289} \][/tex]
[tex]\[ \cos^2 \theta = \frac{225}{289} \][/tex]
Therefore:
[tex]\[ \cos \theta = \sqrt{\frac{225}{289}} \][/tex]
[tex]\[ \cos \theta = \frac{15}{17} \][/tex]
The calculated numerical value for [tex]\(\cos \theta\)[/tex] is approximately [tex]\(0.8824\)[/tex].
2. Calculating [tex]\(\tan \theta\)[/tex]:
We use the identity:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
Substituting the given values:
[tex]\[ \tan \theta = \frac{\frac{8}{17}}{\frac{15}{17}} = \frac{8}{15} \][/tex]
The calculated numerical value for [tex]\(\tan \theta\)[/tex] is approximately [tex]\(0.5333\)[/tex].
3. Calculating [tex]\(\frac{\cos \theta + \tan \theta}{\cos 0 - \tan \theta}\)[/tex]:
Note that [tex]\(\cos 0 = 1\)[/tex].
[tex]\[ \frac{\cos \theta + \tan \theta}{\cos 0 - \tan \theta} = \frac{\frac{15}{17} + \frac{8}{15}}{1 - \frac{8}{15}} \][/tex]
Simplifying the expression:
[tex]\[ \frac{\cos \theta + \tan \theta}{\cos 0 - \tan \theta} = \frac{0.8824 + 0.5333}{1 - 0.5333} \][/tex]
The calculated numerical value for this expression is approximately [tex]\(3.0336\)[/tex].
### Part (b)
Given: [tex]\(\tan \theta = \frac{5}{12}\)[/tex] and [tex]\(\theta\)[/tex] is in the range [tex]\(0^\circ\)[/tex] to [tex]\(90^\circ\)[/tex].
1. Calculating [tex]\(\sin^2 \theta\)[/tex]:
We use the identity:
[tex]\[ \sec^2 \theta = 1 + \tan^2 \theta \][/tex]
Substituting [tex]\(\tan \theta = \frac{5}{12}\)[/tex]:
[tex]\[ \sec^2 \theta = 1 + \left(\frac{5}{12}\right)^2 \][/tex]
[tex]\[ \sec^2 \theta = 1 + \frac{25}{144} \][/tex]
[tex]\[ \sec^2 \theta = \frac{169}{144} \][/tex]
Therefore:
[tex]\[ \sec \theta = \sqrt{\frac{169}{144}} = \frac{13}{12} \][/tex]
Now, calculate [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos \theta = \frac{1}{\sec \theta} = \frac{12}{13} \][/tex]
Using [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]:
[tex]\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{12}{13}\right)^2 \][/tex]
[tex]\[ \sin^2 \theta = 1 - \frac{144}{169} \][/tex]
[tex]\[ \sin^2 \theta = \frac{25}{169} \][/tex]
The calculated numerical value for [tex]\(\sin^2 \theta\)[/tex] is approximately [tex]\(0.1479\)[/tex].
2. Calculating [tex]\(\cos \theta - \sin \theta\)[/tex]:
Here, we've already computed:
[tex]\[ \cos \theta = \frac{12}{13} \][/tex]
And using [tex]\(\sin \theta = \sqrt{\frac{25}{169}} = \frac{5}{13}\)[/tex]:
[tex]\[ \cos \theta - \sin \theta = \frac{12}{13} - \frac{5}{13} = \frac{7}{13} \][/tex]
The calculated numerical value for [tex]\(\cos \theta - \sin \theta\)[/tex] is approximately [tex]\(0.5385\)[/tex].
Therefore, putting everything together:
- [tex]\(\cos \theta \approx 0.8824\)[/tex]
- [tex]\(\tan \theta \approx 0.5333\)[/tex]
- [tex]\(\frac{\cos \theta + \tan \theta}{\cos 0 - \tan \theta} \approx 3.0336\)[/tex]
- [tex]\(\sin^2 \theta \approx 0.1479\)[/tex]
- [tex]\(\cos \theta - \sin \theta \approx 0.5385\)[/tex]
These results provide the detailed step-by-step solution to the given question.
### Part (a)
Given: [tex]\(\sin \theta = \frac{8}{17}\)[/tex]
1. Calculating [tex]\(\cos \theta\)[/tex]:
We use the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substituting [tex]\(\sin \theta = \frac{8}{17}\)[/tex]:
[tex]\[ \left(\frac{8}{17}\right)^2 + \cos^2 \theta = 1 \][/tex]
[tex]\[ \frac{64}{289} + \cos^2 \theta = 1 \][/tex]
[tex]\[ \cos^2 \theta = 1 - \frac{64}{289} \][/tex]
[tex]\[ \cos^2 \theta = \frac{289}{289} - \frac{64}{289} \][/tex]
[tex]\[ \cos^2 \theta = \frac{225}{289} \][/tex]
Therefore:
[tex]\[ \cos \theta = \sqrt{\frac{225}{289}} \][/tex]
[tex]\[ \cos \theta = \frac{15}{17} \][/tex]
The calculated numerical value for [tex]\(\cos \theta\)[/tex] is approximately [tex]\(0.8824\)[/tex].
2. Calculating [tex]\(\tan \theta\)[/tex]:
We use the identity:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
Substituting the given values:
[tex]\[ \tan \theta = \frac{\frac{8}{17}}{\frac{15}{17}} = \frac{8}{15} \][/tex]
The calculated numerical value for [tex]\(\tan \theta\)[/tex] is approximately [tex]\(0.5333\)[/tex].
3. Calculating [tex]\(\frac{\cos \theta + \tan \theta}{\cos 0 - \tan \theta}\)[/tex]:
Note that [tex]\(\cos 0 = 1\)[/tex].
[tex]\[ \frac{\cos \theta + \tan \theta}{\cos 0 - \tan \theta} = \frac{\frac{15}{17} + \frac{8}{15}}{1 - \frac{8}{15}} \][/tex]
Simplifying the expression:
[tex]\[ \frac{\cos \theta + \tan \theta}{\cos 0 - \tan \theta} = \frac{0.8824 + 0.5333}{1 - 0.5333} \][/tex]
The calculated numerical value for this expression is approximately [tex]\(3.0336\)[/tex].
### Part (b)
Given: [tex]\(\tan \theta = \frac{5}{12}\)[/tex] and [tex]\(\theta\)[/tex] is in the range [tex]\(0^\circ\)[/tex] to [tex]\(90^\circ\)[/tex].
1. Calculating [tex]\(\sin^2 \theta\)[/tex]:
We use the identity:
[tex]\[ \sec^2 \theta = 1 + \tan^2 \theta \][/tex]
Substituting [tex]\(\tan \theta = \frac{5}{12}\)[/tex]:
[tex]\[ \sec^2 \theta = 1 + \left(\frac{5}{12}\right)^2 \][/tex]
[tex]\[ \sec^2 \theta = 1 + \frac{25}{144} \][/tex]
[tex]\[ \sec^2 \theta = \frac{169}{144} \][/tex]
Therefore:
[tex]\[ \sec \theta = \sqrt{\frac{169}{144}} = \frac{13}{12} \][/tex]
Now, calculate [tex]\(\cos \theta\)[/tex]:
[tex]\[ \cos \theta = \frac{1}{\sec \theta} = \frac{12}{13} \][/tex]
Using [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]:
[tex]\[ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(\frac{12}{13}\right)^2 \][/tex]
[tex]\[ \sin^2 \theta = 1 - \frac{144}{169} \][/tex]
[tex]\[ \sin^2 \theta = \frac{25}{169} \][/tex]
The calculated numerical value for [tex]\(\sin^2 \theta\)[/tex] is approximately [tex]\(0.1479\)[/tex].
2. Calculating [tex]\(\cos \theta - \sin \theta\)[/tex]:
Here, we've already computed:
[tex]\[ \cos \theta = \frac{12}{13} \][/tex]
And using [tex]\(\sin \theta = \sqrt{\frac{25}{169}} = \frac{5}{13}\)[/tex]:
[tex]\[ \cos \theta - \sin \theta = \frac{12}{13} - \frac{5}{13} = \frac{7}{13} \][/tex]
The calculated numerical value for [tex]\(\cos \theta - \sin \theta\)[/tex] is approximately [tex]\(0.5385\)[/tex].
Therefore, putting everything together:
- [tex]\(\cos \theta \approx 0.8824\)[/tex]
- [tex]\(\tan \theta \approx 0.5333\)[/tex]
- [tex]\(\frac{\cos \theta + \tan \theta}{\cos 0 - \tan \theta} \approx 3.0336\)[/tex]
- [tex]\(\sin^2 \theta \approx 0.1479\)[/tex]
- [tex]\(\cos \theta - \sin \theta \approx 0.5385\)[/tex]
These results provide the detailed step-by-step solution to the given question.
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