Q.If $\cos x = -\frac{3}{5}$, $x$ lies in the third quadrant, find the values of other five trigonometric functions.

Concept: Trigonometric Functions in Quadrants — sign of each function depends on the quadrant.

Step 1: In the third quadrant, both $\sin x$ and $\cos x$ are negative; $\tan x$ is positive.

Step 2: Given $\cos x = -\frac{3}{5}$. Using $\sin^2 x + \cos^2 x = 1$:

So $\sin x = -\frac{4}{5}$ (negative in third quadrant).

Step 3: Now compute the remaining functions:

The key to solving this problem is understanding which trigonometric functions are positive or negative in each quadrant. This isn't just a memorisation trick — it follows from the definitions of sine, cosine, and tangent on the unit circle.

In the third quadrant ($180^\circ < x < 270^\circ$), both the $x$-coordinate (cosine) and $y$-coordinate (sine) are negative. Since tangent is $\frac{\sin x}{\cos x}$, a negative divided by a negative gives a positive value. So in Q3: only $\tan x$ and its reciprocal $\cot x$ are positive; $\sin x$, $\cos x$, $\sec x$, and $\csc x$ are all negative.

Now let's work through the solution step by step.

1. Start with what's given.

   We know $\cos x = -\frac{3}{5}$ and $x$ lies in the third quadrant. The value $-\frac{3}{5}$ tells us the adjacent side (in a reference triangle) is $3$ and the hypotenuse is $5$, but the negative sign tells us the direction along the $x$-axis.

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

   The fundamental identity is:

Substitute $\cos x = -\frac{3}{5}$:

Taking the square root gives $\sin x = \pm \frac{4}{5}$.

Since $x$ is in the third quadrant, $\sin x$ is negative. Therefore:

3. Find $\tan x$ using the ratio of sine to cosine.

Notice the two negatives cancel, giving a positive value — exactly what we expect in Q3.

4. Find the reciprocal functions. Once we have $\sin x$, $\cos x$, and $\tan x$, the remaining three are just reciprocals:
   • $\csc x = \frac{1}{\sin x} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}$
   • $\sec x = \frac{1}{\cos x} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}$
   • $\cot x = \frac{1}{\tan x} = \frac{1}{\frac{4}{3}} = \frac{3}{4}$

Let's verify the signs one more time. In Q3: $\sin x$ negative, $\cos x$ negative, $\tan x$ positive, $\cot x$ positive, $\sec x$ negative, $\csc x$ negative. All our answers match.