Find the trigonometric form of the complex number 𝑧 represented by the given Argand diagram.
In standard form, a complex number can be graphed using rectangular coordinates 𝑎, 𝑏. In this case, 𝑧 is equal to 𝑎 plus 𝑏 multiplied by 𝑖, where 𝑎 represents the 𝑥-coordinate and 𝑏 represents the 𝑦-coordinate.
Alternatively, the 𝑥-coordinate can represent real number values, while the 𝑦-coordinate represents the imaginary values. You’ve probably noticed, however, that we’ve been given values of 𝑟 and 𝜃 from the polar coordinate system. In this case, we can use the formulae 𝑥 is equal to 𝑟 cos 𝜃 and 𝑦 is equal to 𝑟 sin 𝜃 to convert numbers in the polar plane into their trigonometric form.
Now the first part of this is fairly straightforward. We can define our value of 𝑟 as being four, as we’re given this in the question. But what is 𝜃? In this case, 𝜃 is not the angle given. 𝜃 is actually the angle between the 𝑥-axis and 𝑜𝑧. This is 90 minus 30, which is 60 degrees.
Remember though, 𝜃 should be in radians and not degrees. We can therefore convert 60 degrees into radians by multiplying by 𝜋 over 180. 60 multiplied by 𝜋 over 180 is equal to 𝜋 over three radians. In fact, by considering the unit circle, we realise that this is actually negative 𝜋 over three radians because we travelled in a clockwise direction.
Now that we’ve defined our values of 𝑟 and 𝜃, we can substitute this into the conversion formulae we looked at earlier. 𝑥 is equal to four cos of negative 𝜋 over three, and 𝑦 is equal to four sin of negative 𝜋 over three. Finally, we recall that 𝑧 is equal to 𝑎 plus 𝑏 multiplied by 𝑖, where 𝑎 is the 𝑥-coordinate and 𝑏 is the 𝑦-coordinate. 𝑧 is therefore equal to four cos of negative 𝜋 over three plus four sin of negative 𝜋 over three multiplied by 𝑖. Factorising fully gives us a solution of 𝑧 equals four lots of cos of negative 𝜋 over three add 𝑖 multiplied by sin of negative 𝜋 over three.