The question asks to find the argument $\varphi$ of the complex number $z$, where:

$z = 2 \cos \pi + i \sin \pi$

The complex number $z$ is expressed in trigonometric (polar) form as $z = r(\cos \theta + i \sin \theta)$, where $r$ is the modulus, and $\theta$ is the argument.

First, let's break this down:

1. Modulus: The modulus $r = 2$, given by the expression.

2. Argument: The angle $\theta = \pi$ since both trigonometric functions involve $\pi$:

   • $\cos \pi = -1$
   • $\sin \pi = 0$

Thus, the complex number simplifies to:

$z = 2 \cdot (-1 + 0i) = -2$

Since this is a real number, lying on the negative real axis, its argument is $\pi$.

Therefore, the correct argument $\varphi$ is $\pi$.

Answer:

The correct choice is (c) $\pi$.

Would you like more details or have any questions? Here are five related questions for practice:

1. How is the argument of a complex number generally defined?
2. What is the geometric interpretation of the argument $\varphi$ of a complex number?
3. How would the argument change if the complex number was on the positive real axis?
4. What would be the argument if the complex number was on the negative imaginary axis?
5. How is the polar form of a complex number converted back to rectangular form?

Tip: Remember, the argument of a complex number is the angle it makes with the positive real axis, measured counterclockwise.