QPaperGen

Given:
\[
\sec x = \frac{13}{5}, \quad x \text{ lies in the fourth quadrant}
\]

Since \(\sec x = \frac{1}{\cos x}\):
\[
\cos x = \frac{5}{13}
\]
In the fourth quadrant, cosine is positive, sine is negative.

\(\textbf{Find \(\sin x\):}\) by using
\[
\sin^2 x + \cos^2 x = 1
\]
\[
\sin^2 x = 1 - \cos^2 x = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169}
\]
\[
\sin x = -\frac{12}{13}
\]
(Negative in fourth quadrant)
\[\]

Other trigonometric functions:
\[
\tan x = \frac{\sin x}{\cos x} = \frac{-12/13}{5/13} = -\frac{12}{5}
\]
\[
\csc x = \frac{1}{\sin x} = -\frac{13}{12}
\]
\[
\cot x = \frac{\cos x}{\sin x} = \frac{5/13}{-12/13} = -\frac{5}{12}
\]

Summary Table:
\[
\begin{array}{|c|c|}
\hline
\text{Function} & \text{Value} \\
\hline
\sin x & -\frac{12}{13} \\
\cos x & \frac{5}{13} \\
\tan x & -\frac{12}{5} \\
\sec x & \frac{13}{5} \\
\csc x & -\frac{13}{12} \\
\cot x & -\frac{5}{12} \\
\hline
\end{array}
\]