```
5. Points $A$ and $B$ are the representations in the complex plane of the numbers $z=2-2 \mathrm{i}$ and $w=-1-\sqrt{3} \mathrm{i}$ respectively.
(a) Given that the origin is $\mathrm{O}$, find the angle $A O B$ in radians, expressing your answer in terms of $\pi$.
(b) Calculate the argument of $z w$ in radians, again expressing your answer in terms of $\pi$.
Working:
Answers:
```

Sign in to unlock the answer

Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Identify the complex numbers $z$ and $w$ in the complex plane. $z = 2 - 2i$ and $w = -1 - \sqrt{3}i$

step 2

Calculate the arguments of $z$ and $w$. The argument of $z$ is $\arg(z) = \tan^{-1}\left(\frac{-2}{2}\right) = \tan^{-1}(-1) = -\frac{\pi}{4}$

step 3

The argument of $w$ is $\arg(w) = \tan^{-1}\left(\frac{-\sqrt{3}}{-1}\right) = \tan^{-1}(\sqrt{3}) = -\frac{\pi}{3}$

step 4

Find the angle $AOB$ by taking the difference of the arguments: $\theta = \arg(w) - \arg(z) = -\frac{\pi}{3} - (-\frac{\pi}{4}) = -\frac{\pi}{3} + \frac{\pi}{4} = -\frac{4\pi}{12} + \frac{3\pi}{12} = -\frac{\pi}{12}$

```
Part (b)
```

step 1

Calculate the product $zw$. $zw = (2 - 2i)(-1 - \sqrt{3}i)$

step 2

Expand the product: $zw = 2(-1) + 2(-\sqrt{3}i) - 2i(-1) - 2i(-\sqrt{3}i) = -2 - 2\sqrt{3}i + 2i + 2\sqrt{3} = -2 + 2\sqrt{3} - 2\sqrt{3}i + 2i$

step 3

Simplify the expression: $zw = -2 + 2\sqrt{3} - 2\sqrt{3}i + 2i$

step 4

Calculate the argument of $zw$. The argument of $zw$ is $\arg(zw) = \tan^{-1}\left(\frac{2i - 2\sqrt{3}i}{-2 + 2\sqrt{3}}\right) = \tan^{-1}\left(\frac{2i(1 - \sqrt{3})}{2(1 - \sqrt{3})}\right) = \tan^{-1}(i) = \frac{\pi}{2}$

Answer

(a) The angle $AOB$ is $-\frac{\pi}{12}$ radians.

(b) The argument of $zw$ is $\frac{\pi}{2}$ radians.

Key Concept

Complex numbers and their arguments in the complex plane

Explanation

The argument of a complex number is the angle it makes with the positive real axis, and the angle between two complex numbers can be found by taking the difference of their arguments.

Not the question you are looking for? Ask here!

Enter question by text

Enter question by image

Upgrade to Asksia Pro

Join a AskSia's Pro Plan, and get 24/7 AI tutoring for your reviews, assignments, quizzes and exam preps.

Unlimited chat query usages

Strong algorithms that better know you

Early access to new release features

Study Other Question