Complex Numbers Dunman High 2020 P1 Q3 | A Level H2 Mathematics

Hi everyone! Welcome back to another practice question on complex numbers, this is taken from the 2020 Dunman High Paper 1, Question #3.

Timestamps
00:12 Introduction and Question Requirements
00:23 Representing Complex Numbers on Argand Diagram
00:55 How to express arg (w-z) on Argand Diagram
02:07 When do we use the Trigonometric Form of a Complex Number
02:34 Converting Trigonometric Form to Cartesian Form
03:56 Using Sine Rule
04:25 Another example - Converting Trigonometric Form to Cartesian Form

Now this question is not exactly a common one that you usually would see, we are tasked to use an Argand diagram to show that angle OWZ is pi/12 radians. Before we do this, let’s try to represent complex numbers z and w on the diagram with whatever information that we are given.

Since z has an argument of -pi/3, z will be in the 2nd quadrant, as it is measured from the positive x axis, in a clockwise direction.
w has an argument of positive pi/4, so we place it in the first quadrant, anti-clockwise from the x axis, with a modulus of 2.

The argument of (w-z) - when represented on the Argand diagram is actually the vector ZW. Recall that if you want to find vector AB for example, it is position vector OB minus position vector OA. So it’s the same for this, for vector ZW, it is denoted by arg (w-z). And we can denote the argument as the angle that ZW makes with the positive x axis over here, which is given to us as pi/3.

So very simply, we can find angle OWZ by taking pi - pi/4 - 2pi/3 and this will be pi/12. Don’t forget to write the word shown!

In the next part, our objective is to find z and w in the cartesian form. Now since the Imaginary and real components are not given to you directly, not to worry, what we can do is to employ the TRIGONOMETRIC form of z and w, to help us convert into the respective cartesian forms. The trigonometric form is also used because we have the modulus and the angles that are readily available to us!

So for w, we can express in trigonometric form by writing w = 2 (cos pi/4 + i sin pi/4). Simplifying we have sqrt2 (1+i). Now this is not the final answer because we need to express a and b to 3 sig fig. Always answer to the question. Thus w = 1.41+1.41i

To find z in the form of a+bi, we must first find the modulus of z, denoted by OZ. The question has hinted to us that we can make use of angle OWZ, so maybe let’s try to use the sine rule to get OZ. With OZ as the modulus, express z in the trigonometric form and hence, express it in the cartesian form.

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