Question

If $\tan \theta =-\frac{1}{\sqrt{10}}$ and $\theta$ lies in the fourth quadrant,then $\cos \theta =$

• A. $\frac{1}{\sqrt{11}}$
• B. $-\frac{1}{\sqrt{11}}$
• C. $\sqrt{11}$
• D. $\sqrt{\frac{10}{11}}$

Answer 1

The correct option is D $\sqrt{\frac{10}{11}}$

Given:$\tan \theta =-\frac{1}{\sqrt{10}}$ and $\theta$ lies in the fourth quadrant.

$\therefore x$ axis is positive and $y-$axis is negative.

$\Rightarrow \tan \theta =\frac{Oppositeside}{adjacentside}=-\frac{1}{\sqrt{10}}$

Hence $Oppositeside=1$ unit along negative $y-$axis$=-1$

and $adjacentside=\sqrt{10}$ units along positive $x-$axis$=\sqrt{10}$

Hence by Pythagoras theorem,${\left(Oppositeside\right)}^2+{\left(adjacentside\right)}^2={\left(hypotenuse\right)}^2$

$\Rightarrow {(-1)}^2+{\left(\sqrt{10}\right)}^2={\left(hypotenuse\right)}^2$

$\therefore {\left(hypotenuse\right)}^2=1+10=11$

$\therefore hypotenuse=\sqrt{11}$units.

$\therefore \cos \theta =\frac{adjacentside}{hypotenuse}=\frac{\sqrt{10}}{\sqrt{11}}=\sqrt{\frac{10}{11}}$ units