If cot α = 1/2, sec β = -5/3, where π < α < 3π/2 and π/2 < β < π, find the value of tan(α + β). State the quadrant in which α + β terminates.

Question

If cot α = \(\frac{1}{2}\), sec β = \(\frac{-5}{3}\), where π < α < \(\frac{3\pi}{2}\) and \(\frac{\pi}{2}\) < β < π, find the value of tan(α + β). State the quadrant in which α + β terminates.

Answer 1

Given that cot α = \(\frac{1}{2}\) where π < α < \(\frac{3\pi}{2}\)(i.e,. α lies in third quadrant)

tan α = \(\frac{1}{\frac{1}{2}}\)= 2 [∵ In 3rd quadrant tan α is positive] 

Also given that sec β = \(\frac{-5}{3}\) where \(\frac{\pi}{2}\) < β < π (i.e., β lies in second quadrant cos β and tan β are negative)

tan (α + β) = \(\frac{2}{11}\) which is positive.

α + β terminates in first quandrant.