Question

If $\cos \left(A\right)=\frac{4}{5}$ then the value of $\tan \left(A\right)$ is

• A. $\frac{3}{5}$
• B. $\frac{3}{4}$
• C. $\frac{4}{3}$
• D. $\frac{5}{3}$

Answer 1

The correct option is B

$\frac{3}{4}$

Explanation for correct option:

Find the required trigonometric value

Given, $\cos \left(A\right)=\frac{4}{5}$

So,

$$\begin{gathered} \tan \left(A\right)=\frac{\sin \left(A\right)}{\cos \left(A\right)} \\ =\frac{\sqrt{1-{\cos }^2\left(A\right)}}{\cos \left(A\right)}[\because {\sin }^2\left(A\right)+{\cos }^2\left(A\right)=1\Rightarrow \sin \left(A\right)=\sqrt{1-{\cos }^2\left(A\right)}] \\ =\frac{\sqrt{1-{\left({\displaystyle \frac{4}{5}}\right)}^2}}{{\displaystyle \frac{4}{5}}}[\because \cos \left(A\right)=\frac{4}{5}] \\ =\frac{\sqrt{1-{\displaystyle \frac{16}{25}}}}{{\displaystyle \frac{4}{5}}} \\ =\frac{\sqrt{{\displaystyle \frac{25-16}{25}}}}{{\displaystyle \frac{4}{5}}} \\ =\frac{\sqrt{{\displaystyle \frac{9}{25}}}}{{\displaystyle \frac{4}{5}}} \\ =\frac{{\displaystyle \frac{3}{5}}}{{\displaystyle \frac{4}{5}}} \\ =\frac{3}{4} \end{gathered}$$

Hence, $\tan \left(A\right)=\frac{3}{4}$, so, the correct option is (B).