Question

Given $\tan A=\frac{4}{3}$, find all other trigonometric ratios of A.

Answer 1

Finding all the other trigonometric ratios of A.

$$\begin{gathered} \tan A=\frac{4}{3} \\ \tan A=\frac{p}{b}=\frac{4}{3} \end{gathered}$$

$$\begin{gathered} \mathrm{Here},p=4andb=3 \\ h=? \end{gathered}$$

By Pythagoras theorem,

$$\begin{gathered} h^2=p^2+b^2 \\ \Rightarrow h=\sqrt{p^2+b^2} \\ \Rightarrow h=\sqrt{{\left(4\right)}^2+{\left(3\right)}^2} \\ \Rightarrow h=\sqrt{16+9}=\sqrt{25}=5 \\ \therefore h=5 \\ \mathrm{Now}, \\ \sin A=\frac{p}{h}=\frac{4}{5} \\ \cos A=\frac{b}{h}=\frac{3}{5} \\ \cos ecA=\frac{1}{\sin A}=\frac{5}{4} \\ secA=\frac{1}{\cos A}=\frac{5}{3} \\ cotA=\frac{1}{\tan A}=\frac{3}{4} \end{gathered}$$

Hence, other trigonometric ratios are:

$$\begin{gathered} \sin A=\frac{4}{5} \\ \cos A=\frac{3}{5} \\ \cos ecA=\frac{5}{4} \\ secA=\frac{5}{3} \\ cotA=\frac{3}{4} \end{gathered}$$