Question

If $\sec \, \theta - \tan \, \theta = \frac{1}{2},$ then $\theta$ lies in

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

Answer 1

Answer: (A)

the first quadrant

Answer 2

Given $\sec \, \theta - \tan \, \theta = \frac{1}{2}$ ....(1) Also $\sec^2 \, \theta - \tan^2 \, \theta = 1$ ....(2) Dividing (2) by (1), we get $\sec \, \theta + \tan \, \theta = 2$ ....(3) Adding (1) and (3), we get $2 \, \sec \, \theta = \frac{5}{2} \, \Rightarrow \, \sec \, \theta = \frac{5}{4}$ and subtracting (2) from (1), we get $2 \tan \, \theta = \frac{3}{2} \, \Rightarrow \, \tan \, \theta = \frac{3}{4}$ Since both sec $\theta$ and tan $\theta$ are positive, therefore, $\theta$ lies in the first quadrant.