To solve for [tex]\(\cot \theta\)[/tex] given that the angle [tex]\(\theta\)[/tex] lies in the range [tex]\(0^\circ < \theta < 90^\circ\)[/tex] and [tex]\(\cos \theta = \frac{11}{61}\)[/tex], we follow these steps using trigonometric identities:
1. Identify the components of the right triangle:
- We interpret [tex]\(\cos \theta = \frac{11}{61}\)[/tex] based on the definition of cosine in a right triangle:
[tex]\[
\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}.
\][/tex]
This tells us:
[tex]\[
\text{adjacent} = 11 \quad \text{and} \quad \text{hypotenuse} = 61.
\][/tex]
2. Find the opposite side of the triangle using the Pythagorean theorem:
- The Pythagorean theorem states:
[tex]\[
\text{opposite}^2 + \text{adjacent}^2 = \text{hypotenuse}^2.
\][/tex]
Substituting the known values:
[tex]\[
\text{opposite}^2 + 11^2 = 61^2.
\][/tex]
[tex]\[
\text{opposite}^2 + 121 = 3721.
\][/tex]
Solving for [tex]\(\text{opposite}^2\)[/tex]:
[tex]\[
\text{opposite}^2 = 3721 - 121 = 3600,
\][/tex]
hence:
[tex]\[
\text{opposite} = \sqrt{3600} = 60.
\][/tex]
3. Determine [tex]\(\cot \theta\)[/tex]:
- We use the definition of cotangent:
[tex]\[
\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}.
\][/tex]
With the values we have determined:
[tex]\[
\cot \theta = \frac{11}{60}.
\][/tex]
Therefore, the correct value of [tex]\(\cot \theta\)[/tex] given [tex]\(\cos \theta = \frac{11}{61}\)[/tex] is:
[tex]\[
\frac{11}{60}.
\][/tex]
The correct option is:
[tex]\[
\frac{11}{60}.
\][/tex]
