Question

Find the exact value of csc θ if cotangent theta equals negative 2 and the terminal side of θ lies in quadrant II.
(1 point)
Responses

cosecant theta equals start root 5 end root
Image with alt text: cosecant theta equals start root 5 end root

cosecant theta equals negative start root 5 end root
Image with alt text: cosecant theta equals negative start root 5 end root

cosecant theta equals negative start fraction start root 5 end root over 2 end fraction
Image with alt text: cosecant theta equals negative start fraction start root 5 end root over 2 end fraction

cosecant theta equals start fraction start root 5 end root over 2 end fraction

Answer 1

To find the exact value of \(\csc \theta\) given that \(\cot \theta = -2\) and \(\theta\) is in quadrant II, we can follow these steps:

1. Recall the definition of cotangent:
\[
\cot \theta = \frac{\cos \theta}{\sin \theta}
\]
Since \(\cot \theta = -2\), we have:
\[
\frac{\cos \theta}{\sin \theta} = -2 \implies \cos \theta = -2 \sin \theta
\]

2. In quadrant II, sine is positive and cosine is negative. Let \(\sin \theta = y\) and \(\cos \theta = x\). Then:
\[
x = -2y
\]

3. Use the Pythagorean identity:
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
Substituting for \(x\):
\[
y^2 + (-2y)^2 = 1 \implies y^2 + 4y^2 = 1 \implies 5y^2 = 1 \implies y^2 = \frac{1}{5} \implies y = \frac{1}{\sqrt{5}} \text{ (positive in quadrant II)}
\]

4. Calculate \(\cos \theta\):
\[
\cos \theta = -2y = -2 \cdot \frac{1}{\sqrt{5}} = -\frac{2}{\sqrt{5}}
\]

5. Now, we can find \(\csc \theta\):
\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{y} = \sqrt{5}
\]

Thus, the exact value of \(\csc \theta\) is:
\[
\csc \theta = \sqrt{5}
\]

The correct response is:
\[
\text{cosecant theta equals } \sqrt{5}
\]