# Question Video: Finding the Rectangular and Polar Forms of Complex Numbers on Argand Diagrams

The Argand diagram shows the complex number 𝑧. Write 𝑧 in rectangular form. Convert 𝑧 to polar form, rounding the argument to two decimal places.

02:25

### Video Transcript

The Argand diagram shows the complex number 𝑧. Write 𝑧 in rectangular form.

When we write a complex number 𝑧 in rectangular form, we write it as 𝑎 plus 𝑏𝑖. 𝑎 is the real component and 𝑏 is the imaginary component. And on our diagram, the horizontal axis is the real axis and the vertical axis is the imaginary axis. This means that the values of 𝑎 and 𝑏 can be read straight from the graph, a little like reading Cartesian coordinates. 𝑎 is therefore three and 𝑏 is five. So in rectangular form, the complex number is three plus five 𝑖.

Convert 𝑧 to polar form, rounding the argument to two decimal places.

When we write a complex number in polar or trigonometric form, we write it as 𝑧 equals 𝑟 multiplied by cos 𝜃 plus 𝑖 sin 𝜃. 𝑟 is the modulus of the complex number 𝑧, and 𝜃 is the argument. In polar form, 𝜃 can be in degrees and radians, though radians is often preferred, whereas in exponential form, it does need to be in radians. So we need to find a way to represent the real and complex components of our number 𝑧 in terms of 𝑟 and 𝜃.

In fact, we can use these formulae to help us. The modulus, 𝑟, is the square root of 𝑎 squared plus 𝑏 squared. This comes from the Pythagorean theorem. To find 𝜃, we use tan of 𝜃 is equal to 𝑏 over 𝑎.

Let’s substitute what we know about our complex number into these formulae. 𝑎 is three and 𝑏 is five. So the modulus is the square root of three squared plus five squared, which is root 34. Tan 𝜃 is equal to five-thirds. And we can solve this to find the value of 𝜃 or find the argument by finding the arc tangent of five-thirds. That’s 1.0303, and so on. Correct to two decimal places as required in the question, that’s 1.03 radians.

Now that we have the values of 𝑟 and 𝜃, we can substitute them into the formula for the polar or trigonometric form of the complex number. In polar form, this complex number can be written as root 34 multiplied by cos of 1.03 plus 𝑖 sin of 1.03.