Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive 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]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.