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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]
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