Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Discover solutions to your questions from experienced professionals across multiple fields on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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]
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]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.
