To find the values of \(\sin \theta\) and \(\tan \theta\) given that \(\cot \theta = \frac{1}{8}\) and assuming that \(\theta\) is an acute angle, follow these steps:
1. Understand the given information:
- \(\cot \theta = \frac{1}{8}\).
2. Relate cotangent to tangent:
- We know that \(\cot \theta = \frac{1}{\tan \theta}\).
- Therefore, \(\tan \theta = \frac{1}{\cot \theta} = \frac{1}{\frac{1}{8}} = 8\).
3. Use the Pythagorean Identity:
- The Pythagorean identity states that \(\sin^2 \theta + \cos^2 \theta = 1\).
- We can also express \(\tan \theta\) in terms of sine and cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
4. Relate tangent and cosine:
- We know \(\tan \theta = 8\).
- Thus, \(\sin \theta = 8 \cos \theta\).
5. Substitute into Pythagorean Identity:
- Substitute \(\sin \theta = 8 \cos \theta\) into \(\sin^2 \theta + \cos^2 \theta = 1\):
[tex]\[
(8 \cos \theta)^2 + \cos^2 \theta = 1
\][/tex]
[tex]\[
64 \cos^2 \theta + \cos^2 \theta = 1
\][/tex]
[tex]\[
65 \cos^2 \theta = 1
\][/tex]
- Solve for \(\cos \theta\):
[tex]\[
\cos^2 \theta = \frac{1}{65}
\][/tex]
[tex]\[
\cos \theta = \sqrt{\frac{1}{65}} \approx 0.1240347346
\][/tex]
6. Find \(\sin \theta\):
- Using \(\sin \theta = 8 \cos \theta\):
[tex]\[
\sin \theta = 8 \times 0.1240347346 \approx 0.9922778767
\][/tex]
So, the values are:
- \(\sin \theta \approx 0.9922778767\)
- \(\tan \theta = 8\)
Thus, [tex]\(\sin \theta \approx 0.9922778767\)[/tex] and [tex]\(\tan \theta = 8\)[/tex] for the given condition [tex]\(\cot \theta = \frac{1}{8}\)[/tex].
