Question

If tan $\mathrm{\theta }=-\frac{1}{\sqrt{5}}$ and θ lies in the IV quadrant, then the value of cos θ is
(a) $\frac{\sqrt{5}}{\sqrt{6}}$
(b) $\frac{2}{\sqrt{6}}$
(c) $\frac{1}{2}$
(d) $\frac{1}{\sqrt{6}}$

Answer 1

(a) $\frac{\sqrt{5}}{\sqrt{6}}$

$$\begin{gathered} \mathrm{In}\mathrm{the}\mathrm{fourth}\mathrm{quadrant},\cos \theta \mathrm{and}\sec \theta \mathrm{are}\mathrm{positive}. \\ \cos \theta =\frac{1}{sec\theta } \\ =\frac{1}{\sqrt{se{c}^2\theta }} \\ =\frac{1}{\sqrt{1+{\tan }^2\theta }} \\ =\frac{1}{\sqrt{1+{\left(-{\displaystyle \frac{1}{\sqrt{5}}}\right)}^2}} \\ =\frac{1}{\sqrt{{\displaystyle \frac{6}{5}}}} \\ =\frac{\sqrt{5}}{\sqrt{6}} \end{gathered}$$