Question:
If $\sin \theta = \frac{1}{5}$ and the angle $\theta$ is in the second quadrant, then $\sec \theta$ is equal to:
If $\sin \theta = \frac{1}{5}$ and the angle $\theta$ is in the second quadrant, then $\sec \theta$ is equal to:
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"All Silver Tea Cups" helps remember which functions are positive in each quadrant (S for Sine is positive in the 2nd).
Updated On: Apr 28, 2026
- $\frac{5}{2\sqrt{6}}$
- $\frac{-2\sqrt{6}}{5}$
- $\frac{2\sqrt{6}}{5}$
- $\frac{\sqrt{6}}{5}$
- $\frac{-5}{2\sqrt{6}}$
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The Correct Option is
Solution and Explanation
Step 1: Concept
In the second quadrant, $\cos \theta$ and $\sec \theta$ are negative.
Step 2: Analysis
$\cos^2 \theta = 1 - \sin^2 \theta = 1 - (\frac{1}{5})^2 = 1 - \frac{1}{25} = \frac{24}{25}$. $\cos \theta = -\sqrt{\frac{24}{25}} = -\frac{2\sqrt{6}}{5}$ (negative in quadrant II).
Step 3: Calculation
$\sec \theta = \frac{1}{\cos \theta} = -\frac{5}{2\sqrt{6}}$. Final Answer: (E)
In the second quadrant, $\cos \theta$ and $\sec \theta$ are negative.
Step 2: Analysis
$\cos^2 \theta = 1 - \sin^2 \theta = 1 - (\frac{1}{5})^2 = 1 - \frac{1}{25} = \frac{24}{25}$. $\cos \theta = -\sqrt{\frac{24}{25}} = -\frac{2\sqrt{6}}{5}$ (negative in quadrant II).
Step 3: Calculation
$\sec \theta = \frac{1}{\cos \theta} = -\frac{5}{2\sqrt{6}}$. Final Answer: (E)
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