Certainly! Let's solve the problem step by step.
We start with the given information that \(\cos \theta = \frac{15}{17}\).
### Step 1: Recall the Pythagorean Identity
The Pythagorean Identity states:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
### Step 2: Substitute \(\cos \theta\) into the identity
We have \(\cos \theta = \frac{15}{17}\). Therefore,
[tex]\[ \cos^2 \theta = \left(\frac{15}{17}\right)^2 = \frac{225}{289} \][/tex]
### Step 3: Solve for \(\sin^2 \theta\)
Using the identity:
[tex]\[ \sin^2 \theta = 1 - \cos^2 \theta \][/tex]
Substitute \(\cos^2 \theta\):
[tex]\[ \sin^2 \theta = 1 - \frac{225}{289} \][/tex]
To subtract these fractions, we use a common denominator of 289:
[tex]\[ \sin^2 \theta = \frac{289}{289} - \frac{225}{289} = \frac{64}{289} \][/tex]
### Step 4: Solve for \(\sin \theta\)
Since \(\theta\) is an acute angle, \(\sin \theta\) will be positive:
[tex]\[ \sin \theta = \sqrt{\frac{64}{289}} = \frac{8}{17} \][/tex]
### Step 5: Determine \(\csc \theta\)
The cosecant of \(\theta\) is the reciprocal of the sine of \(\theta\):
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
Substitute \(\sin \theta\):
[tex]\[ \csc \theta = \frac{1}{\frac{8}{17}} = \frac{17}{8} \][/tex]
Thus, the value of \(\csc \theta\) is \(\frac{17}{8}\).
The correct answer is:
[tex]\[
\boxed{\frac{17}{8}}
\][/tex]
