Mathematics

The key is using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to find $\sin x$, then leveraging the quadrant rule to pick the correct sign. Since $x$ is in the third quadrant, both sine and cosine are negative—this constraint eliminates ambiguity when you solve for $\sin x$. Once you have both, you can find any other trig ratio like $\tan x$ or $\sec x$ by definition.

The key is using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to find $\sin x$, then leveraging the quadrant rule to pick the correct sign. Since $x$ is in the third quadrant, both sine and cosine are negative—this constraint eliminates ambiguity when you solve for $\sin x$. Once you have both, you can find any other trig ratio like $\tan x$ or $\sec x$ by definition.