Question

If $\tan \theta =\frac{3}{4}$, then find the value of $\sec \theta$.

Answer 1

$\tan \theta =\frac{3}{4}$

$\Rightarrow \frac{\sin \theta }{\cos \theta }=\frac{3}{4}$

$\Rightarrow 4\cdot \sin \theta =3\cdot \cos \theta$

$\Rightarrow 4^2.(1-{\cos }^2\theta )=3^2{\cos }^2\theta$

$\Rightarrow 16-16{\cos }^2\theta =9{\cos }^2\theta$

$\Rightarrow 25{\cos }^2\theta =16$

$\Rightarrow {\cos }^2\theta =16/25$

$\Rightarrow \cos \theta =\sqrt{16/25}$

$\Rightarrow \cos \theta =4/5$

$\sec \theta =1/\cos \theta$

$=\frac{1}{4/5}$

$=\frac{5}{4}$

$\therefore \sec \theta =\frac{5}{4}$.