To solve the problem where [tex]\(\sin \theta = \frac{\sqrt{75}}{10}\)[/tex] and [tex]\(\cot \theta\)[/tex] is negative, follow these steps:
1. Find [tex]\(\theta\)[/tex] using the given [tex]\(\sin \theta\)[/tex]:
Given [tex]\(\sin \theta = \frac{\sqrt{75}}{10}\)[/tex]. This simplifies to [tex]\(\sin \theta = \frac{5\sqrt{3}}{10} = \frac{\sqrt{3}}{2}\)[/tex].
2. Determine the reference angle:
The reference angle [tex]\(\theta_{\text{ref}}\)[/tex] which we will denote as [tex]\(\theta_{\text{ref}} = \arcsin \left( \frac{\sqrt{3}}{2} \right)\)[/tex]. The angle in degrees for which [tex]\(\sin\)[/tex] is [tex]\(\frac{\sqrt{3}}{2}\)[/tex] is [tex]\(60^\circ\)[/tex].
3. Determine the quadrants where [tex]\(\cot \theta\)[/tex] is negative:
Since [tex]\(\cot \theta\)[/tex] (which is [tex]\(\frac{\cos \theta}{\sin \theta}\)[/tex]) is negative, [tex]\(\theta\)[/tex] must be in either the second or fourth quadrants because:
- In the first quadrant, both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are positive so [tex]\(\cot \theta\)[/tex] would be positive.
- In the second quadrant, [tex]\(\sin \theta\)[/tex] is positive and [tex]\(\cos \theta\)[/tex] is negative resulting in a negative [tex]\(\cot \theta\)[/tex].
- In the third quadrant, [tex]\(\sin \theta\)[/tex] is negative and [tex]\(\cos \theta\)[/tex] is negative resulting in a positive [tex]\(\cot \theta\)[/tex].
- In the fourth quadrant, [tex]\(\sin \theta\)[/tex] is negative and [tex]\(\cos \theta\)[/tex] is positive resulting in a negative [tex]\(\cot \theta\)[/tex].
4. Adjust the reference angle to the correct quadrants:
- For the second quadrant: [tex]\(\theta = 180^\circ - \theta_{\text{ref}}\)[/tex]
[tex]\[
\theta = 180^\circ - 60^\circ = 120^\circ
\][/tex]
- For the fourth quadrant: [tex]\(\theta = 360^\circ - \theta_{\text{ref}}\)[/tex]
[tex]\[
\theta = 360^\circ - 60^\circ = 300^\circ
\][/tex]
5. Choose the angle where [tex]\(\cot \theta\)[/tex] is negative:
We analyze the values:
- For [tex]\(\theta = 120^\circ\)[/tex]:
[tex]\[
\cot(120^\circ) = \frac{\cos(120^\circ)}{\sin(120^\circ)} = \frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \approx -0.5774
\][/tex]
- For [tex]\(\theta = 300^\circ\)[/tex]:
[tex]\[
\cot(300^\circ) = \frac{\cos(300^\circ)}{\sin(300^\circ)} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \approx -0.5774
\][/tex]
Both values give us a negative [tex]\(\cot \theta\)[/tex], but since the angle [tex]\(\theta\)[/tex] should be primary (one within the stated interval that usually appears first in trigonometric calculations for an angle), we select:
[tex]\(\theta = 120^\circ\)[/tex].
Thus, the exact answers are:
[tex]\[
\theta = 120^\circ \quad \text{and} \quad \cot \theta = -\frac{\sqrt{3}}{3}
\][/tex]
