If =1 and = -53, where

Question

If $\cot\alpha=1$ and $\sec\beta = -\dfrac53$, where $\pi<\alpha<\dfrac{3\pi}2$ and $\dfrac\pi2<\beta<\pi$, then the value of $	an(\alpha+\beta)$ and the quadrant in which $\alpha+\beta$ lies, respectively, are:

Accepted Answer

Since $\cot\alpha=1$ with $\alpha$ in the third quadrant, $\tan\alpha=1$.

Since $\sec\beta=-\tfrac53$ with $\beta$ in the second quadrant, $\cos\beta=-\tfrac35$, $\sin\beta=\tfrac45$, so $\tan\beta=-\tfrac43$.

$$\tan(\alpha+\beta) = \frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta} = \frac{1-\tfrac43}{1+\tfrac43} = \frac{-1/3}{7/3} = -\frac17$$

Since $\alpha\in\left(\pi,\tfrac{3\pi}2\right)$ and $\beta\in\left(\tfrac\pi2,\pi\right)$, $\alpha+\beta\in\left(\tfrac{3\pi}2,2\pi\right)$ --- the fourth quadrant.

Solutions · 1