To find [tex]\(\cot \theta\)[/tex] given [tex]\(\cos \theta = \frac{11}{61}\)[/tex] for [tex]\(0^\circ < \theta < 90^\circ\)[/tex], follow these steps:
1. Understand the identity involving sine and cosine:
We know the Pythagorean identity for trigonometric functions:
[tex]\[
\cos^2 \theta + \sin^2 \theta = 1
\][/tex]
2. Substitute [tex]\(\cos \theta\)[/tex] into the identity:
Given [tex]\(\cos \theta = \frac{11}{61}\)[/tex], we square this value to find [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\cos^2 \theta = \left(\frac{11}{61}\right)^2 = \frac{121}{3721}
\][/tex]
3. Solve for [tex]\(\sin^2 \theta\)[/tex]:
Using the identity [tex]\(\cos^2 \theta + \sin^2 \theta = 1\)[/tex]:
[tex]\[
\sin^2 \theta = 1 - \cos^2 \theta = 1 - \frac{121}{3721} = \frac{3721 - 121}{3721} = \frac{3600}{3721}
\][/tex]
4. Calculate [tex]\(\sin \theta\)[/tex]:
Since we are considering [tex]\(0^\circ < \theta < 90^\circ\)[/tex], [tex]\(\sin \theta\)[/tex] must be positive:
[tex]\[
\sin \theta = \sqrt{\frac{3600}{3721}} = \frac{60}{61}
\][/tex]
5. Use the definition of [tex]\(\cot \theta\)[/tex]:
We know that [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]. Substitute the values we found for [tex]\(\cos \theta\)[/tex] and [tex]\(\sin \theta\)[/tex]:
[tex]\[
\cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{11}{61}}{\frac{60}{61}} = \frac{11}{60}
\][/tex]
Thus, the value of [tex]\(\cot \theta\)[/tex] is [tex]\(\boxed{\frac{11}{60}}\)[/tex].
