Find the values of other five trigonometric functions if cot x = 3/4, x lies in the third quadrant.

Sure, let's solve this step by step.

Given that cot x = 3/4, we know that tan x = 1/cot x = 4/3.

Since x is in the third quadrant where both sine and cosine are negative, tan x is positive.

Now, we can find the other trigonometric functions.

1. sin x = -sqrt(1 - cos²x)
2. cos x = -sqrt(1 - sin²x)
3. sec x = 1/cos x
4. csc x = 1/sin x
5. cot x = 1/tan x

We can use the Pythagorean identity sin²x + cos²x = 1 to find sin x and cos x.

From tan x = sin x / cos x, we can express sin x and cos x in terms of tan x:

sin x = tan x / sqrt(1 + tan²x) = 4/5 cos x = 1 / sqrt(1 + tan²x) = -3/5

Then, we can find the other trigonometric functions:

sec x = 1/cos x = -5/3 csc x = 1/sin x = 5/4 cot x = 1/tan x = 3/4

So, the values of the six trigonometric functions are:

sin x = -4/5 cos x = -3/5 tan x = 4/3 csc x = -5/4 sec x = -5/3 cot x = 3/4