Q.Find the values of other five trigonometric functions if $\sec x = \frac{13}{5}$, $x$ lies in fourth quadrant.

Concept: Trigonometric functions in quadrants using reciprocal and Pythagorean identities.

Since $\sec x = \frac{13}{5}$, we have $\cos x = \frac{5}{13}$ (positive, consistent with quadrant IV).

Using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$:

So $\sin x = \pm\frac{12}{13}$. In quadrant IV, sine is negative: $\sin x = -\frac{12}{13}$.

Now the remaining functions follow:

$\tan x = \frac{\sin x}{\cos x} = \frac{-12/13}{5/13} = -\frac{12}{5}$ …

The secant function tells us the ratio of hypotenuse to adjacent side in a right triangle. When we know one trigonometric function and the quadrant, we can reconstruct the entire picture because all six functions are interconnected through identities and sign conventions.

In the fourth quadrant, the reference angle sits below the positive $x$-axis. Here, $x$-coordinates are positive and $y$-coordinates are negative. This means $\cos x > 0$ and $\sin x < 0$, which cascades to the signs of all other functions.

Since $\sec x = \frac{1}{\cos x} = \frac{13}{5}$, we immediately have:

Now we'll find the remaining functions systematically.

1. Find $\sin x$ using the Pythagorean identity

The fundamental identity $\sin^2 x + \cos^2 x = 1$ gives us:

Taking the square root: $|\sin x| = \frac{12}{13}$

Since $x$ is in the fourth quadrant where sine is negative:

2. Find $\tan x$ from sine and cosine

The tangent is the ratio of sine to cosine:

3. Find the reciprocal functions

The cosecant is the reciprocal of sine:

The cotangent is the reciprocal of tangent:

$\cot x = \frac{1}{\tan x} = \frac{1}{-\frac{12}{5}} = -\frac{5}{12}$ …