Question

$\mathrm{If}\mathrm{sin\theta }=\frac{3}{5},\mathrm{where}\theta \in \left(0,\frac{\pi }{2}\right),\mathrm{then}({\sec }^2\theta +{\tan }^2\theta )\mathrm{is}17k.$
then the value of k is .

• A. $\frac{1}{8}$
• B. $\frac{1}{16}$
• C. $\frac{1}{32}$

Answer 1

The correct option is A $\frac{1}{8}$

Given $\sin \theta =\frac{3}{5}=\frac{\left(\mathrm{opposite}\right)}{\left(\mathrm{hypotenuse}\right)}\Rightarrow \mathrm{adjacent}\mathrm{side}=\sqrt{5^2-3^2}=4.$
Now, we can write
$sec\theta =\frac{5}{4}\mathrm{and}\tan \theta =\frac{3}{4}$.
$se{c}^2\theta +{\tan }^2\theta =\frac{25}{16}+\frac{9}{16}\Rightarrow se{c}^2\theta +{\tan }^2\theta =\frac{34}{16}=\frac{17}{8}.$