Question

find the sin x and tan x if cos c =-12/13,where x lies in the third quadrant

Answer 1

Given:

$cos(x)=\frac{-12}{13}$, where $x$ lies in the third quadrant.

To find:

$sin(x)$ and $tan(x)$

Solution:

$sin(x)$ is negative and $tan(x)$ is positive in the third quadrant, since $x$ lies in the third quadrant.

We know that $cos(x^{2} )+sin(x^{2} )=1$

$sin^{2} (x )=1-cos^{2} (x )\\sin(x)=\sqrt{1-cos^{2} (x )} \\sin(x)=\sqrt{1-(\frac{-12}{13} )^{2} } \\sin(x)=\sqrt{1-\frac{144}{169} }=\sqrt{\frac{25}{169} } \\sin(x)=\frac{5}{13}$

But $sin(x)$ is negative in the third quadrant,

∴$sin(x)=\frac{-5}{13}$

Next, we know that $tan(x)=\frac{sin(x)}{cos(x)}$

$tan(x)=\frac{\frac{-5}{13} }{\frac{-12}{13} }$

⇒$tan(x)=\frac{5}{12}$

Hence, $sin(x)=\frac{-5}{13}$ and $tan(x)=\frac{5}{12}$.