To solve for [tex]\(\tan \theta\)[/tex] given that [tex]\(\sin \theta = \frac{9}{41}\)[/tex], we should follow these steps.
### Step 1: Understand Trigonometric Identities
We will use the basic trigonometric identities - specifically the Pythagorean identity and the definition of tangent:
1. [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]
2. [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex]
### Step 2: Use the Pythagorean Identity
Given [tex]\(\sin \theta = \frac{9}{41}\)[/tex], we first find [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substitute [tex]\(\sin \theta\)[/tex] into this identity:
[tex]\[
\left(\frac{9}{41}\right)^2 + \cos^2 \theta = 1
\][/tex]
Calculate [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\left(\frac{9}{41}\right)^2 = \frac{81}{1681}
\][/tex]
Now solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\frac{81}{1681} + \cos^2 \theta = 1
\][/tex]
[tex]\[
\cos^2 \theta = 1 - \frac{81}{1681}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{1681}{1681} - \frac{81}{1681}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{1600}{1681}
\][/tex]
### Step 3: Find [tex]\(\cos \theta\)[/tex]
[tex]\[
\cos \theta = \sqrt{\frac{1600}{1681}}
\][/tex]
Since [tex]\(\theta\)[/tex] is typically taken within a range where cosine is positive:
[tex]\[
\cos \theta = \frac{\sqrt{1600}}{\sqrt{1681}}
\][/tex]
[tex]\[
\cos \theta = \frac{40}{41}
\][/tex]
### Step 4: Use the Definition of Tangent
Now, we know both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sin \theta = \frac{9}{41}
\][/tex]
[tex]\[
\cos \theta = \frac{40}{41}
\][/tex]
We can find [tex]\(\tan \theta\)[/tex] using its definition:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
Substitute the known values:
[tex]\[
\tan \theta = \frac{\frac{9}{41}}{\frac{40}{41}}
\][/tex]
Simplify the fraction division:
[tex]\[
\tan \theta = \frac{9}{40}
\][/tex]
### Conclusion
The correct answer is:
[tex]\[
\tan \theta = \frac{9}{40}
\][/tex]
Thus, the answer is:
[tex]\[
\boxed{\frac{9}{40}}
\][/tex]
This matches choice C in the given options.
