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!