Question Video: Sketching Regions That the Complex Number Satisfies in the Complex Plane

Video Transcript

Sketch on an Argand diagram the region represented by the argument of 𝑧 plus three minus two 𝑖 is greater than or equal to negative 𝜋 by two and less than 𝜋 by four.

To sketch this region, we’ll begin by considering the boundaries. They are given by the argument of 𝑧 plus three minus two 𝑖 is equal to negative 𝜋 by two and the argument of 𝑧 plus three minus two 𝑖 is equal to 𝜋 by four. Each of these represents a half line. We can rewrite 𝑧 plus three minus two 𝑖 by factoring negative one. And we get 𝑧 minus negative three plus two 𝑖. The point that represents this complex number will have Cartesian coordinates negative three, two. And of course, we represent this with an open circle since we know that the locus of points doesn’t actually include this point.

The first boundary is going to make an angle of negative 𝜋 by two with the positive horizontal, measured in a counterclockwise direction. This is the same as measuring an angle of positive 𝜋 by two in the clockwise direction. And this is a weak inequality. So we draw a solid line for this one as shown. The half line for our next boundary will make an angle of 𝜋 by four radians with the positive horizontal, measured in a counterclockwise direction. This time, it’s a weak inequality. So we need to draw a dashed line as shown.

Now that we have the boundaries for our region, we need to decide which side of the region we’re going to shade. We’re interested in all the complex numbers such that the argument of 𝑧 plus three minus two 𝑖 is greater than or equal to negative 𝜋 by two and less than 𝜋 by four. That’s going to be the region that lies between these two half lines. So we shade this region. And we’re done. We’ve sketched the required region on an Argand diagram.