Let's solve it step-by-step.
We start with the given information:
[tex]\[ \cos \theta = \frac{24}{25} \][/tex]
We need to find the value of [tex]\(\csc \theta\)[/tex], which is the reciprocal of [tex]\(\sin \theta\)[/tex].
First, let's determine [tex]\(\sin \theta\)[/tex] using the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Since [tex]\(\cos \theta = \frac{24}{25}\)[/tex]:
[tex]\[ \sin^2 \theta + \left(\frac{24}{25}\right)^2 = 1 \][/tex]
We solve for [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[ \sin^2 \theta + \frac{576}{625} = 1 \][/tex]
Subtract [tex]\(\frac{576}{625}\)[/tex] from both sides:
[tex]\[ \sin^2 \theta = 1 - \frac{576}{625} \][/tex]
[tex]\[ \sin^2 \theta = \frac{625}{625} - \frac{576}{625} \][/tex]
[tex]\[ \sin^2 \theta = \frac{49}{625} \][/tex]
Take the square root of both sides to find [tex]\(\sin \theta\)[/tex]:
[tex]\[ \sin \theta = \sqrt{\frac{49}{625}} \][/tex]
[tex]\[ \sin \theta = \frac{7}{25} \][/tex]
Now, we find [tex]\(\csc \theta\)[/tex], which is the reciprocal of [tex]\(\sin \theta\)[/tex]:
[tex]\[ \csc \theta = \frac{1}{\sin \theta} \][/tex]
[tex]\[ \csc \theta = \frac{1}{\frac{7}{25}} \][/tex]
[tex]\[ \csc \theta = \frac{25}{7} \][/tex]
Thus, the value of [tex]\(\csc \theta \)[/tex] is:
[tex]\[ \boxed{\frac{25}{7}} \][/tex]
