IDNLearn.com is the place where your questions are met with thoughtful and precise answers. Get the information you need from our experts, who provide reliable and detailed answers to all your questions.
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.
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
