Question

Find the value of other five trigonometric functions in cos x = - $\frac{1}{2}$, x lies in third quadrant.

Answer 1

Here cos x = - $\frac{1}{2}$

Now $si{n}^{2}x+co{s}^2$ x =1

$\Rightarrow si{n}^{2}x+{(-\frac{1}{2})}^2=1$

$\Rightarrow si{n}^{2}x=1-\frac{1}{4}$

$\Rightarrow si{n}^{2}x=\frac{3}{4}$

$\Rightarrow sinx=\pm \frac{\sqrt{3}}{2}$

But x lies in third quadrant.

$\therefore$ sin x =- $\frac{\sqrt{3}}{2}$

Now cosec x = $\frac{1}{sinx}=-\frac{2}{\sqrt{3}}$

and sec x = $\frac{1}{\text{cos x}}=-2$

tan x = $\frac{\text{sin x}}{\text{cos x}}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}$

and cot x =$\frac{\text{cos x}}{\text{sin x}}=\frac{\frac{-\sqrt{1}}{2}}{\frac{-\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}$