Certainly! Let's go through the problem step-by-step, given that [tex]\(\sin \theta = \frac{15}{17}\)[/tex].
### Step 1: Understand the given value
We know that [tex]\(\sin \theta = \frac{15}{17}\)[/tex].
This implies:
- The opposite side to angle [tex]\(\theta\)[/tex] (in a right triangle) is 15 units.
- The hypotenuse of the triangle is 17 units.
### Step 2: Find [tex]\(\cos \theta\)[/tex]
Using the Pythagorean identity, [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex],
[tex]\[
\cos \theta = \sqrt{1 - \sin^2 \theta}
\][/tex]
[tex]\[
\cos \theta = \sqrt{1 - \left(\frac{15}{17}\right)^2}
\][/tex]
[tex]\[
\cos \theta = \sqrt{1 - \frac{225}{289}}
\][/tex]
[tex]\[
\cos \theta = \sqrt{\frac{289 - 225}{289}}
\][/tex]
[tex]\[
\cos \theta = \sqrt{\frac{64}{289}}
\][/tex]
[tex]\[
\cos \theta = \frac{8}{17}
\][/tex]
### Step 3: Verify each option
Option A: [tex]\(\sec \theta = \frac{17}{8}\)[/tex]
Recall that [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex],
[tex]\[
\sec \theta = \frac{1}{\frac{8}{17}} = \frac{17}{8}
\][/tex]
This is correct.
Option B: [tex]\(\csc \theta = \frac{17}{15}\)[/tex]
Recall that [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex],
[tex]\[
\csc \theta = \frac{1}{\frac{15}{17}} = \frac{17}{15}
\][/tex]
This is correct.
Option C: [tex]\(\tan \theta = \frac{15}{8}\)[/tex]
Recall that [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex],
[tex]\[
\tan \theta = \frac{\frac{15}{17}}{\frac{8}{17}} = \frac{15}{8}
\][/tex]
This is correct.
Option D: [tex]\(\cos \theta = \frac{17}{8}\)[/tex]
We already calculated [tex]\(\cos \theta\)[/tex] above and found that,
[tex]\[
\cos \theta = \frac{8}{17}
\][/tex]
So, [tex]\(\cos \theta = \frac{17}{8}\)[/tex] is incorrect.
### Conclusion
Based on the calculations, the correct answers are:
- A: [tex]\(\sec \theta = \frac{17}{8}\)[/tex]
- B: [tex]\(\csc \theta = \frac{17}{15}\)[/tex]
- C: [tex]\(\tan \theta = \frac{15}{8}\)[/tex]
Therefore, the correct options are A, B, and C.
