Video Transcript
Find the trigonometric form of the
complex number 𝑍 represented by the given Argand diagram.
We begin by recalling that the
trigonometric form of a complex number is written 𝑍 is equal to 𝑟 multiplied by
cos 𝜃 plus 𝑖 sin 𝜃, where 𝑟 is the magnitude or length of the complex number and
𝜃 is its argument. From the diagram, we can see that
𝑟, the magnitude of 𝑍, is equal to four. Finding the value of 𝜃, however,
is more complicated. It is the angle that 𝑍 makes with
the positive real axis. We recall that angles measured in
the counterclockwise direction are positive, and angles measured in the clockwise
direction are negative. This means that we could add the
angles from zero to 360 degrees onto our diagram such that 𝑍 lies at an angle of
300 degrees in the positive direction. Likewise, if we consider the
negative direction, we see that 𝑍 lies at an angle of negative 60 degrees from the
positive real axis.
We now have two possible values of
𝜃. However, the argument of any
complex number must be written in radians such that 𝜃 is greater than negative 𝜋
and less than or equal to 𝜋. Recalling that 𝜋 radians is equal
to 180 degrees, we can add 𝜋 over two, 𝜋, and negative 𝜋 over two to our
diagram. Dividing through by three, we see
that 𝜋 over three radians is equal to 60 degrees, and as such, 𝜃 — the argument of
𝑍 — is equal to negative 𝜋 over three.
The trigonometric form of the
complex number 𝑍 represented by the Argand diagram is 𝑍 is equal to four
multiplied by cos of negative 𝜋 over three plus 𝑖 sin negative 𝜋 over three.
