Question

$\mathrm{If}\sec \theta +\tan \theta =\frac{1}{2},\mathrm{then}\text{'}\theta \text{'}\mathrm{lies}\mathrm{in}\mathrm{the}$

• A. first quadrant.
• B. second quadrant.
• C. third quadrant.
• D. fourth quadrant.

Answer 1

The correct option is D fourth quadrant.

$\mathrm{Given},\sec \theta +\tan \theta =\frac{1}{2}\mathrm{we}\mathrm{know}\mathrm{that}{\sec }^2\theta -{\tan }^2\theta =1\Rightarrow (\sec \theta +\tan \theta )(\sec \theta -\tan \theta )=1\Rightarrow \frac{1}{2}(\sec \theta -\tan \theta )=1\Rightarrow \sec \theta -\tan \theta =2\mathrm{Now},\mathrm{adding}\mathrm{both}\mathrm{equations}\mathrm{we}\mathrm{get},2\sec \theta =\frac{5}{2}\Rightarrow \sec \theta =\frac{5}{4}\Rightarrow \cos \theta =\frac{4}{5}\mathrm{Now},\mathrm{substituting}\mathrm{the}\mathrm{value}\mathrm{of}\sec \theta ,\mathrm{we}\mathrm{get}\tan \theta =\frac{-3}{4}\mathrm{Now},\mathrm{looking}\mathrm{at}\mathrm{the}\mathrm{signs}\mathrm{of}\mathrm{both}\cos \theta \mathrm{and}\tan \theta \mathrm{we}\mathrm{see}\cos \theta >0\mathrm{and}\tan \theta <0\mathrm{Thus},\text{'}\theta \text{'}\mathrm{lies}\mathrm{in}\mathrm{the}\mathrm{fouth}\mathrm{quadrant}.$