Find the angle of complex numbers

The  exercise appears under the Precalculus Math Mission. This exercise introduces the concept of the absolute value, or magnitude, for complex numbers.

Types of Problems
There is one type of problem in this exercise:


 * 1) Find the angle $$\theta$$ (in radians) that $$z$$ makes in the complex plane.: This problem has an absolute value equation and the student is expected to find the angle that $$z$$ makes in the complex plane. Find the angle of complex numbers.PNG

Strategies
Knowledge of the Pythagorean theorem and radical simplification are encouraged to ensure success on this exercise.
 * 1) Students can find the angle $$\theta$$ of any complex number $$z$$ by solving the following equation: $$tan\theta = \frac{\text{Lm}(z)}{\text{Re}(z)}$$
 * 2) This equation usually has two solutions in the interval $$[-\pi, \pi]$$. Users can find the appropriate solution by reasoning about the quadrant in which $$z$$ lies.

Real-life Applications

 * 1) Complex numbers have applications as models in non-Euclidean geometries.
 * 2) Complex numbers are used to create fractal images.
 * 3) Complex numbers are a natural extension of the real numbers.