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.
