At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Connect with a community of experts ready to help you find solutions to your questions quickly and accurately. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine the true statements given [tex]\(\cos \theta = \frac{15}{17}\)[/tex], we need to first calculate [tex]\(\sin \theta\)[/tex], and then use it to find the values of [tex]\(\tan \theta\)[/tex], [tex]\(\csc \theta\)[/tex], and [tex]\(\sec \theta\)[/tex].
1. Calculate [tex]\(\sin \theta\)[/tex] using the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substituting [tex]\(\cos \theta = \frac{15}{17}\)[/tex], we get:
[tex]\[ \sin^2 \theta + \left(\frac{15}{17}\right)^2 = 1 \][/tex]
[tex]\[ \sin^2 \theta + \frac{225}{289} = 1 \][/tex]
[tex]\[ \sin^2 \theta = 1 - \frac{225}{289} \][/tex]
[tex]\[ \sin^2 \theta = \frac{289}{289} - \frac{225}{289} \][/tex]
[tex]\[ \sin^2 \theta = \frac{64}{289} \][/tex]
[tex]\[ \sin \theta = \sqrt{\frac{64}{289}} \][/tex]
[tex]\[ \sin \theta = \frac{8}{17} \][/tex]
2. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
[tex]\[ \tan \theta = \frac{\frac{8}{17}}{\frac{15}{17}} \][/tex]
[tex]\[ \tan \theta = \frac{8}{15} \][/tex]
3. Calculate [tex]\(\csc \theta\)[/tex] (the reciprocal of [tex]\(\sin \theta\)[/tex]):
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
[tex]\[ \csc \theta = \frac{1}{\frac{8}{17}} \][/tex]
[tex]\[ \csc \theta = \frac{17}{8} \][/tex]
4. Calculate [tex]\(\sec \theta\)[/tex] (the reciprocal of [tex]\(\cos \theta\)[/tex]):
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
[tex]\[ \sec \theta = \frac{1}{\frac{15}{17}} \][/tex]
[tex]\[ \sec \theta = \frac{17}{15} \][/tex]
Now we can check each statement:
- Statement A: [tex]\(\tan \theta = \frac{8}{15}\)[/tex]
This is True.
- Statement B: [tex]\(\sin \theta = \frac{15}{8}\)[/tex]
This is False. The correct [tex]\(\sin \theta\)[/tex] is [tex]\(\frac{8}{17}\)[/tex].
- Statement C: [tex]\(\csc \theta = \frac{17}{15}\)[/tex]
This is False. The correct [tex]\(\csc \theta\)[/tex] is [tex]\(\frac{17}{8}\)[/tex].
- Statement D: [tex]\(\sec \theta = \frac{17}{15}\)[/tex]
This is True.
So the verified answers are:
- A. [tex]\(\tan \theta = \frac{8}{15}\)[/tex]
- D. [tex]\(\sec \theta = \frac{17}{15}\)[/tex]
1. Calculate [tex]\(\sin \theta\)[/tex] using the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substituting [tex]\(\cos \theta = \frac{15}{17}\)[/tex], we get:
[tex]\[ \sin^2 \theta + \left(\frac{15}{17}\right)^2 = 1 \][/tex]
[tex]\[ \sin^2 \theta + \frac{225}{289} = 1 \][/tex]
[tex]\[ \sin^2 \theta = 1 - \frac{225}{289} \][/tex]
[tex]\[ \sin^2 \theta = \frac{289}{289} - \frac{225}{289} \][/tex]
[tex]\[ \sin^2 \theta = \frac{64}{289} \][/tex]
[tex]\[ \sin \theta = \sqrt{\frac{64}{289}} \][/tex]
[tex]\[ \sin \theta = \frac{8}{17} \][/tex]
2. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
[tex]\[ \tan \theta = \frac{\frac{8}{17}}{\frac{15}{17}} \][/tex]
[tex]\[ \tan \theta = \frac{8}{15} \][/tex]
3. Calculate [tex]\(\csc \theta\)[/tex] (the reciprocal of [tex]\(\sin \theta\)[/tex]):
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
[tex]\[ \csc \theta = \frac{1}{\frac{8}{17}} \][/tex]
[tex]\[ \csc \theta = \frac{17}{8} \][/tex]
4. Calculate [tex]\(\sec \theta\)[/tex] (the reciprocal of [tex]\(\cos \theta\)[/tex]):
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
[tex]\[ \sec \theta = \frac{1}{\frac{15}{17}} \][/tex]
[tex]\[ \sec \theta = \frac{17}{15} \][/tex]
Now we can check each statement:
- Statement A: [tex]\(\tan \theta = \frac{8}{15}\)[/tex]
This is True.
- Statement B: [tex]\(\sin \theta = \frac{15}{8}\)[/tex]
This is False. The correct [tex]\(\sin \theta\)[/tex] is [tex]\(\frac{8}{17}\)[/tex].
- Statement C: [tex]\(\csc \theta = \frac{17}{15}\)[/tex]
This is False. The correct [tex]\(\csc \theta\)[/tex] is [tex]\(\frac{17}{8}\)[/tex].
- Statement D: [tex]\(\sec \theta = \frac{17}{15}\)[/tex]
This is True.
So the verified answers are:
- A. [tex]\(\tan \theta = \frac{8}{15}\)[/tex]
- D. [tex]\(\sec \theta = \frac{17}{15}\)[/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
