r) 4cos π 3 cos 2π + 3ctg(-)-2sin(-)

To simplify the expression \( r = 4 \cos \frac{\pi}{3} \) and \( \cos 2\pi + 3 \cot(-) - 2\sin(-) \), let's break it down step by step.

Step 1: Calculate \( 4 \cos \frac{\pi}{3} \)

First, we need to find \( \cos \frac{\pi}{3} \): \[ \cos \frac{\pi}{3} = \frac{1}{2} \] Now, multiply by 4: \[ 4 \cos \frac{\pi}{3} = 4 \times \frac{1}{2} = 2 \]

Step 2: Calculate \( \cos 2\pi \)

Now, calculate \( \cos 2\pi \): \[ \cos 2\pi = 1 \]

Step 3: Evaluate the terms involving \( \cot(-) \) and \( \sin(-) \)

Without specific values for the angles (denoted by \(-\)), we can't simplify \( \cot(-) \) and \( \sin(-) \) any further. Recall that:

• The cotangent function \(\cot x\) is defined as \(\cot x = \frac{\cos x}{\sin x}\).
• The sine function \(\sin(-x) = -\sin x\).

If we assume the common negative angle \( x \) represents an angle, we can express:

• \( \cot(-x) = -\cot x \)
• \( \sin(-x) = -\sin x \)

So we rewrite \( 3 \cot(-) - 2 \sin(-) \): \[ = 3(-\cot x) - 2(-\sin x) = -3 \cot x + 2 \sin x \] Without specific values for \(-\), we cannot further simplify this.

Final Expression

The final simplified expression combining the parts we can calculate is: \[ r = 2 + \cos 2\pi + 3\cot(-) - 2\sin(-) = 2 + 1 - 3\cot x + 2\sin x \] So we have: \[ r = 3 - 3\cot(-) + 2\sin(-) \]

This means we can combine results to finalize our expression as follows: \[ r = 3 - 3 \cot(-) + 2 \sin(-) \]

Feel free to provide any specific values for \( - \) to get a numerical result if needed!