### Interesting Trigonometry Question

We know how A’levels like to combine a few topics across. We also know how bad trigonometry can be. I was coaching my International Bachelorette (IB) class last week and came across a fairly interesting question. It tests students on their abilities to manage double angle identities. Not a tough question, but definitely good practice 🙂
FYI: All double angle identities are found in MF26.
Here it is…

(a) Show that $(1 + i \text{tan} \theta)^n + (1 - i \text{tan} \theta)^n = \frac{2 \text{cos} n \theta}{ \text{cos}^n \theta}$, $\text{cos} \theta \neq 0$.

(b)
(i) Use the double angle identity $\text{tan} 2 \theta = \frac{2 \text{tan} \theta}{1 - \text{tan}^2 \theta}$ to show that $\text{tan} \frac{\pi}{8} = \sqrt{2} - 1$.

(ii) Show that $\text{cos} 4x = 8 \text{cos}^4 x - 8 \text{cos}^2 x + 1$.

(iii) Hence find the value of $\int^{\frac{\pi}{8}}_0 {\frac{2 \text{cos} 4x}{ \text{cos}^2 x}}~dx$ exactly.

### Random Questions from 2017 Prelims #1

Last year, I shared a handful of random interesting questions from the 2016 Prelims. Students feedback that they were quite helpful and gave them good exposure. I thought I share some that I’ve seen this year. I know, its a bit early for Prelims. But ACJC just had their paper 1. 🙂

This is from ACJC 2017 Prelims Paper 1 Question 7. And it is on complex numbers.

7
(a) Given that $2z + 1 = |w|$ and $2w-z = 4+8i$, solve for $w$ and $z$.

(b) Find the exact values of $x$ and $y$, where $x, y \in \mathbb{R}$ such that $2e^{-(\frac{3+x+iy}{i})} = 1 -i$

I’ll put the solutions up if I’m free.

But for students stuck, consider checking this link here for (a) and this link here for (b). These links hopefully enlightens students.

Just FYI, you cannot $\text{ln}$ complex numbers as they are not real…

### Thinking [email protected] #8

[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be found here

This is a interesting Complex Number Question.

The complex number $z$ satisfies $z + |z| = 2 + 8i$. What is $|z|^2$

### A little history of e

Some students remarked on why I actually recognise e, that is, $e=2.718281828...$. Well, e is a rather unique constants. Firstly, for all JC students, we see it our daily algebra & complex numbers. Students exposed to university statistics will see e appearing in the formula for normal distribution, that is, $f(x | \mu , \sigma^2) = \frac{1}{\sqrt{2 \sigma^2 \pi}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}$.

Secondly, the story of how it came about is pretty cool as you will observed in the video below.

The Story of e

Hopefully it provides you with another perspective towards this constants! And now you should be more cautious when signing up savings plans that give interest per annum or per month.

### Thoughts about 2016 A’levels H2 Mathematics Paper

I’ve covered some things in classes, with sufficient revisions and final lap papers set. So I thought we have a little breakdown. And of course, we should review what was weeded away in the 9740 H2 Mathematics Syllabus. After all, this is the very LAST time the can test them.

1. Recurrence Relations. I’ve harped on this last year too. Conjectures! Conjectures! You can read more about it here. An a question involving conjecture should start with a recurrence relation, then a conjecture, ending with a recurrence MI. Students should know how to do both $\Sigma$ and recurrence MI. Yes, they are different.

2. Loci. You guys are the lucky last batch to do Loci. So please buy a protractor and compass. Draw them, as one of my student put it, surgically. If need be, use a graph paper (why not?). Harder loci for example can require students to draw for example, $\text{arg}(z-1+2i) = \text{tan}^{-1}(\frac{4}{3})$. You should not have trouble measuring this angle, because you could not even be able to do it. Students should be able to draw such angles with ease. One little note about Loci, will definitely their geometrical descriptions. Many students can draw this, but stumble to describe them.

3. Vectors. Truth be told, I’m still waiting for a question involving vectors in 3D, to land in A’levels. An example can be the HCI Prelims Paper 1 Question 6, which can be found here.

4. Poisson Distribution. I don’t know what this topic has been axed. So students should ready for one big Poisson Distribution questions, I say give it 12-14 marks. And it should be tested with conditional probability. I’ll practice either Demand & Supply or Inflow & Outflow questions. An example can be the one found in NYJC Prelims Paper 2 Question 11, which can be found here.

5. Correlation & Regression. Ever wondered what the $r^2$ means in the GC? Well, $r^2 = bd$ is being removed from the syllabus as well. It hasn’t surfaced before, so maybe it shall finally make its one and only LAST presence felt this year. Students should familiarise themselves with the use of $y- \bar{y} = b ( x - \bar{x})$ equation, which can be found in the MF15. I know many of you probably have not seen it before.

6. Hypothesis Testing. Students should review definitions of level of significance and p-value. Also understand what you may conclude from a Z-Test, using the results of a T-test. A little small part that students can think about, is why use a small sample size? After all, we know that have a sufficiently large $n$ allows us to perform CLT and then use a Z-Test.

7. Trigonometry. After it appeared in 2011 for a trigonometry MI, the product to sum formulas is still a problem for most students. I highly doubt its coming out again with MI, but its can easily come out again with complex numbers. An example can be this.

More examples and discussion will be made in class.

### Trigonometry used in complex Numbers

Given $z = \text{cos}\alpha + i \text{sin} \alpha$ and $w = \text{cos}\beta + i \text{sin} \beta$

$z - w = \text{cos}\alpha + i \text{sin} \alpha -(\text{cos}\beta + i \text{sin} \beta)$

$z - w = \text{cos}\alpha -\text{cos}\beta + + i \text{sin} \alpha - i \text{sin} \beta)$

$z - w = - 2 \text{sin}(\frac{\alpha-\beta}{2})\text{sin}(\frac{\alpha+\beta}{2}) + i 2 \text{sin}(\frac{\alpha-\beta}{2})\text{cos}(\frac{\alpha+\beta}{2})$

$z - w = 2 i \text{sin}(\frac{\alpha-\beta}{2}) (\text{cos}(\frac{\alpha+\beta}{2}) + i \text{sin}(\frac{\alpha+\beta}{2}))$

$z - w = 2 i \text{sin}(\frac{\alpha-\beta}{2}) e^{\frac{\alpha + \beta}{2}}$

### Random Questions from 2016 Prelims #11

TPJC/P2/4

(a) The complex number w is given by $3+3\sqrt{3}i$
(i) Find the modulus and argument of w, giving your answer in exact form.
(ii) Without using a calculator, find the smallest positive integer value of n for which $(\frac{w^3}{w^*})^n$ is a real number.

(b) The complex number z is such that $z^5 = - 4 \sqrt{2}$
(i) Find the value of z in the form $re^{i\theta}$, where $r > 0$ and $- \pi \textless \theta \le \pi$.
(ii) Show the roots on an argand diagram.
(iii) The roots represented by $z_1$ and $z_2$ are such that $0 \textless arg({z_1}) \textless arg({z_2}) \textless \pi$. The locus of all points z such that $|z - z_1| = |z-z_2|$ intersects the line segment joining points representing $z_1$ and $z_2$ at the point P. P represents the complex number p. Find, in exact form, the modulus and argument of p.

### Random Questions from 2016 Prelims #10

PJC P2 Q1

The complex numbers a and b are given by $2 + 3i$ and $-4-5i$ respectively.

(i) On a single Argand diagram, sketch the loci
(a) $|2z-a-b| = |a-b|$
(b) $0 \le \text{arg}(z-b) \le \text{arg}(a - b)$

(ii) Find the range of $\text{arg}(z)$ where $z$ is the complex number that satisfies the relationships in part (i)

### Random Questions from 2016 Prelims #6

HCI P1 Q5

Sketch on a single Argand diagram, the loci defined by $-\frac{\pi}{4} \textless \text{arg}(z+1+2i) \le \frac{\pi}{4}$ and $|(2+i)w+5| \le \sqrt{5}$

(i) Find the minimum value of $\text{arg}(w)$

(ii) Find the minimum value of $|z-w|$

(iii) Given that $\text{arg}(z-w) \textless \theta, - \pi \textless \theta \le \pi$, state the minimum value of $\theta$

### Golden Nugget!!!

$1 + 2 + 3 + 4 + 5 + ... = -\frac{1}{12}$

My brother shared this interesting video with me a few days back when he was at the screening for The Man Who Knew Infinity. I’m looking forward to watching this movie too!

Back to the video! It focuses on the sum that is written above. And interestingly, this sum that should not be defined (as what JC students learnt in Arithmetic Progression), is actually a NEGATIVE number. Explain excellently by professor Edward Frenkel. He brings in interesting concepts from complex numbers too. Hopefully, this piques some interest!

Professor Edward Frenkel wrote a book “Love & Math”, which is really intriguing. You don’t have to love math to read it but you will after reading 🙂

You can read more here too.