Principal Arguments of z and $-z$ and also $z-w$ and $w-z$.

 A student asked me about these.

Try them before you look at the answer!!

Q3

Q4

Examples of solving complex equations involving arguments

 

Factorise $p(z)=z^5-5z^4+10z^3-10z^2-11z+15$ given that $z=1-2i$ is a root and that three roots are integers.(mobius)

 

This is the factorisation of $p(z)$ into real linear factors and real quadratic factors.

ANSWER: The answer to the original question is that the complex linear factors of $p(z)$ are $$(z-1),(z+1),(z-3),(z-1+2i)(z-1-2i)\ .$$

Find mod and Arg of a power of a complex number in mod-arg form. Example

 

Ch3 Q61 Factorise $z^6+8$ into both linear and quadratic factors.

 

Riemann Sums example Ch8 Q1iii

 

Understanding Polar Coordinates

 

Express $\cos^5t$ in terms of $\cos 5t,\cos3t,\cos t$ and express $\sin^4t$ in terms of $\cos 4t, \cos 2t$

• Other examples like this one are contained in the Algebra Notes Ch3 page 98 Example 3 and 4 ! Please also read this book, as it has many useful examples!!
• RTB - Read the Book 😎

Basic Riemann Sum for $I=\displaystyle\int^5_0x(5-x)\,dx$ with 5 intervals.

  

Riemann Sums (Upper and Lower) for an increasing function $y=f(x)$ on partition $\mathcal{P}=\left\{0,\dfrac{1}{n},\dfrac{2}{n},\cdots, \dfrac{n-1}{n},1\right\}$

  

Polar graph $r=2+\sin\theta$.

  

Polar graph $r=3+\cos\theta$.

  

Polar Graph blank 1

  

Sketch the ray $\text{Arg}\ (z-(3+2i))=\dfrac{\pi}{4}$

  

Find the square roots of $i$ in two ways, Cartesian and Mod-Arg way.