Video Transcript
Consider the diagram. Which of the following correctly
describes the relationship between 𝑎, 𝑟, and 𝜃? Is it a) 𝑎 equals 𝑟 tan 𝜃, b) 𝑎
equals 𝑟 sin 𝜃, c) 𝑎 equals cos 𝜃 divided by 𝑟, d) 𝑎 equals 𝑟 cos 𝜃, or e)
𝑎 equals sin 𝜃 over 𝑟?
𝑧 equals 𝑎 plus 𝑏𝑖 is known as
the rectangular or sometimes algebraic form of the complex number 𝑧. We need to find a way to represent
the real and complex components of this number in terms of 𝑟 and 𝜃, where 𝑟 is
known as the modulus of the complex number and 𝜃 is the argument. Let’s consider the right-angled
triangle with side lengths 𝑎, 𝑏, and 𝑟. We can use right- angled
trigonometry to find an expression for 𝑎 in terms of 𝑟 and 𝜃.
Labelling the triangle as shown, we
can see that to find an expression of 𝑎 in terms of 𝑟 and 𝜃, we’ll need to find
the trigonometric ratio that links the adjacent with the hypotenuse. That’s the cosine ratio; cos 𝜃 is
adjacent divided by hypotenuse. Substitute in what we know about
the dimensions of our triangle into this formula, and we get cos 𝜃 is 𝑎 over
𝑟. To form an equation for 𝑎 in terms
of 𝑟 and 𝜃, let’s multiply both sides by 𝑟. Doing so, we get 𝑟 cos 𝜃 is equal
to 𝑎.
And this tells us the formula that
correctly describes the relationship between 𝑎, 𝑟 and 𝜃. It’s d: 𝑎 equals 𝑟 cos 𝜃.
Which of the following correctly
describes the relationship between 𝑏, 𝑟 and 𝜃? Is it a) 𝑏 equals 𝑟 sin 𝜃, b) 𝑏
equals cos 𝜃 divided by 𝑟, c) 𝑏 equals 𝑟 cos 𝜃, d) 𝑏 equals sin 𝜃 over 𝑟, or
e) 𝑏 equals 𝑟 tan 𝜃?
Let’s repeat the process from
before. This time though, we’re looking to
form an expression for 𝑏 in terms of 𝑟 and 𝜃. 𝑏 is the opposite side in the
triangle. It’s a side opposite the included
angle 𝜃. So we need to find a ratio that
links the opposite side with hypotenuse. That’s the sine ratio; sin 𝜃 is
opposite over hypotenuse. Substituting what we know about the
dimensions of our triangle into this formula, and we get sin 𝜃 is equal to 𝑏 over
𝑟.
Once again we can multiply both
sides of this equation by 𝑟 and we get 𝑟 sin 𝜃 is equal to 𝑏. And this means the formula that
correctly describes the relationship between 𝑏, 𝑟 and 𝜃 is a: 𝑏 equals 𝑟 sin
𝜃.
Hence, express 𝑧 in terms of 𝑟
and 𝜃. We were given our complex number in
rectangular or algebraic form: 𝑧 equals 𝑎 plus 𝑏𝑖. We now have expressions for 𝑎 and
𝑏 in terms of 𝑟 and 𝜃. So let’s substitute these into the
expression for the complex number 𝑧. That’s 𝑧 equals 𝑟 cos 𝜃 plus 𝑟
sin 𝜃 multiplied by 𝑖. We can factorize both 𝑟 cos 𝜃 and
𝑟 sin 𝜃 multiplied by 𝑖 share a common factor of 𝑟.
So we can write 𝑧 as 𝑟 multiplied
by cos 𝜃 plus 𝑖 sin 𝜃. And in doing so, we have expressed
𝑧 in terms of 𝑟 and 𝜃. You might sometimes see this called
trigonometric form or polar form.
