Question

If $\mathrm{tan\theta }=\frac{3}{4}$, find the values of sec​θ and cos​θ

Answer 1

We have,

$$\begin{gathered} {\sec }^2\theta =1+{\tan }^2\theta \\ \Rightarrow {\sec }^2\theta =1+{\left(\frac{3}{4}\right)}^2 \\ \Rightarrow {\sec }^2\theta =1+\frac{9}{16}=\frac{16+9}{16}=\frac{25}{16} \\ \Rightarrow \sec \theta =\sqrt{\frac{25}{16}}=\frac{5}{4} \end{gathered}$$
Now,

$$\begin{gathered} \cos \theta =\frac{1}{\sec \theta } \\ \Rightarrow \cos \theta =\frac{1}{\left({\displaystyle \frac{5}{4}}\right)} \\ \Rightarrow \cos \theta =\frac{4}{5} \end{gathered}$$
Thus, the values of sec​θ and cos​θ are $\frac{5}{4}$ and $\frac{4}{5}$, respectively.