Question

if tan theta =-1/root 5 and theta lies in the 2 quadrant, then the value of cos theta is

Answer 1

tanФ=1/√5, tanФ=sinФ/cosФ
90<Ф<180
cosФ=  adj/hyp         (AC^2=AB^2+BC^2)
         =    √5/√6                   AC^2=1+5=6
        
         

Answer 2

$\bold{\cos \theta \ value \ is -\frac{\sqrt{5}}{\sqrt{6}}}$

Given:

$\tan \theta=-\frac{1}{\sqrt{5}}$

Step-by-step explanation:

Tan trigonometric function is positive in first and third quadrants.

Cos trigonometric function is positive in first and fourth quadrants.

As theta is in second quadrant, tan and cos functions are negative.

$\tan \theta=\frac{\text {Opposite side}}{\text {Adjacent side}}$

$\cos \theta=\frac{\text {Adjacent side}}{\text {Hypotenuse}}$

From Pythagoras theorem,

Square of the hypotenuse is sum of the squares of other two sides.

$A C^{2}=A B^{2}+B C^{2}$

$A C^{2}=1+(\sqrt{5})^{2}=1+5=6$

$A C=\sqrt{6}$

$\cos \theta=\frac{\text {Adjacent side}}{\text {Hypotenuse}}$

Therefore,

$\cos \theta=-\frac{\sqrt{5}}{\sqrt{6}}$