Question

Find the exact value of cot θ if cosecant theta equals negative Start Fraction 4 over 3 End Fraction and and the terminal side of θ lies in Quadrant III.
(1 point)
Responses

cotangent theta equals Start Fraction 4 over 5 End Fraction
Image with alt text: cotangent theta equals Start Fraction 4 over 5 End Fraction

cotangent theta equals Start Fraction 5 over 4 End Fraction
Image with alt text: cotangent theta equals Start Fraction 5 over 4 End Fraction

cotangent theta equals Start Fraction Start Root 7 End Root over 3 End Fraction
Image with alt text: cotangent theta equals Start Fraction Start Root 7 End Root over 3 End Fraction

cotangent theta equals Start Fraction 3 Start Root 7 End Root over 7 End Fraction

Answer 1

To find the exact value of cotangent \( \cot \theta \) given that \( \csc \theta = -\frac{4}{3} \) and that the terminal side of \( \theta \) lies in Quadrant III, we can follow these steps:

1. **Understand the relationship between cosecant and sine**:
\[
\csc \theta = \frac{1}{\sin \theta}
\]
Therefore:
\[
\sin \theta = -\frac{3}{4}
\]
(since cosecant is negative in Quadrant III).

2. **Use the Pythagorean identity** to find cosine:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Substitute \( \sin \theta \):
\[
\left(-\frac{3}{4}\right)^2 + \cos^2 \theta = 1
\]
\[
\frac{9}{16} + \cos^2 \theta = 1
\]
\[
\cos^2 \theta = 1 - \frac{9}{16} = \frac{16}{16} - \frac{9}{16} = \frac{7}{16}
\]
\[
\cos \theta = -\sqrt{\frac{7}{16}} = -\frac{\sqrt{7}}{4}
\]
(since cosine is also negative in Quadrant III).

3. **Find cotangent**:
Cotangent is defined as follows:
\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\]
Substitute the values of \( \cos \theta \) and \( \sin \theta \):
\[
\cot \theta = \frac{-\frac{\sqrt{7}}{4}}{-\frac{3}{4}} = \frac{\sqrt{7}}{3}
\]

Thus, the exact value of \( \cot \theta \) is:
\[
\cot \theta = \frac{\sqrt{7}}{3}
\]

From the given responses, the answer is:
- \( \cot \theta = \frac{\sqrt{7}}{3} \)