Algebra is the metaphysics of arithmetic.

John Ray

These are not letters, they are numbers.
Me, just there

## Introduction: What is$x$??

Consider an equation: a mathematical statement expressing that two objects — e.g. numbers — are equal, equivalent and one and the same. As an example;

$x^2+1=5x$.

In mathematics there are a number of uses for this = sign. There is the common;

$1+2=3$

which merely asserts that the sum of 1 and 2 is the same as 3. Also there is the definition-type :=;

$3^3:=3\times3\times 3$

which defines $3^3$ for example, and by extension all these positive integer powers. Finally there is the equation or formula type =, the most famous of which is probably

Energy=(mass)$\times$ (speed of light) $\times$ (speed of
light)

$E=mc^2$.

An equation of this type is a statement that one object — e.g. a number — is equal to another.

### Introduction

To successfully analyse and solve the equations of Leaving Cert Applied Maths projectiles, one must be very comfortable with trigonometry.

Projectile trigonometry all takes place in $[0^\circ,90^\circ]$ so we should be able to work exclusively in right-angled-triangles (RATs), however I might revert to the unit circle for proofs (without using the unit circle, the definitions for zero and $90^\circ$ are found by using continuity).

Recalling that two triangles are similar if they have the same angles, the fundamental principle governing trigonometry might be put something like this:

Similar triangles differ only by a scale factor.

We show this below, but what this means is that the ratio of corresponding sides of similar triangles are the same, and if one of the angles is a right-angle, it means that if you have an angle, say $40^\circ$, and calculate the ratio of, say, the length of the opposite to the length of the hypotenuse, that your answer doesn’t depend on how large your triangle is and so it makes sense to talk about this ratio for $40^\circ$ rather than just a specific triangle:

These are two similar triangles. The opposite/hypotenuse ratio is the same in both cases.

Suppose the dashed triangle is a $k$-scaled version of the smaller triangle. Then $|A'B'|=k|AB|$ and $|A'C'|=k|AC|$. Thus the opposite to hypotenuse ratio for the larger triangle is

$\displaystyle \frac{|A'B'|}{|A'C'|}=\frac{k|AB|}{k|AC|}=\frac{|AB|}{|AC|}$,

which is the same as the corresponding ratio for the smaller triangle.

This allows us to define some special ratios, the so-called trigonometric ratios. If you are studying Leaving Cert Applied Maths you know what these are. You should also be aware of the inverse trigonometric functions. Also you should be able to, given the hypotenuse and angle, find comfortably the other two sides. We should also know that sine is maximised at $90^\circ$, where it is equal to one.

In projectiles we use another trigonometric ratio:

$\displaystyle \sec(\theta)=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{1}{\cos\theta}$.

Note $\cos90^\circ=0$, so that $\sec90^\circ$ is not defined. Why? Answer here.

### The Pythagoras Identity

For any angle $\theta$,

$\sin^2\theta+\cos^2\theta=1$.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 3

In Week 3 we finished looking at quadratic equations and exponents, and started looking at logarithms.

## Week 4

In Week 4 we will finish looking at logarithms and start the second chapter on Sets and Relations.

## Assessment 1

Test 1 will be held at 09:00, Friday 20 October in Week 6. Note this is different to your assessment schedule provided by the head of department, Tim Horgan. Expect a sample this time next week. Only material from Weeks 1-5 will be examined.

## Study

We are probably all going to have to put in some extra study before the test. Please try and find some extra time to try exercises, particularly with any new or harder material such as logs. We all have two hours of tutorials before the test and finding two extra hours to do exercises will make a big difference. Please feel free to ask me questions about the exercises via email or even better on this webpage.

## Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc. There are some excellent notes on Blackboard for MATH6055.

In this short note we will explain why we multiply matrices in this “rows-by-columns” fashion. This note will only look at $2\times 2$ matrices but it should be clear, particularly by looking at this note, how this generalises to matrices of arbitrary size.

First of all we need some objects. Consider the plane $\Pi$. By fixing an origin, orientation ($x$– and $y$-directions), and scale, each point $P\in\Pi$ can be associated with an ordered pair $(a,b)$, where $a$ is the distance along the $x$ axis and $b$ is the distance along the $y$ axis. For the purposes of linear algebra we denote this point $P=(a,b)$ by

$\displaystyle P=\left(\begin{array}{c}a\\ b\end{array}\right)$.

We have two basic operations with points in the plane. We can add them together and we can scalar multiply them according to, if $Q=(c,d)$ and $\lambda\in\mathbb{R}$:

$P+Q=\left(\begin{array}{c}a\\ b\end{array}\right)+\left(\begin{array}{c}c\\ d\end{array}\right)$

$\displaystyle=\left(\begin{array}{c}a+c\\ b+d\end{array}\right)$, and

$\lambda\cdot P=\lambda\cdot \left(\begin{array}{c}a\\ b\end{array}\right)=\left(\begin{array}{c}\lambda\cdot a\\ \lambda\cdot b\end{array}\right)$.

Objects in mathematics that can be added together and scalar-multiplied are said to be vectorsSets of vectors are known as vector spaces and a feature of vector spaces is that all vectors can be written in a unique way as a sum of basic vectors.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 3

We looked at the applications of vectors to work and moments. We began Chapter 2: Matrices.

## Week 4

We will do some examples of matrix arithmetic and look at Matrix Inverses — “dividing” for Matrices.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 3

In Week 3 we finished Curve Fitting and started the second chapter on Differential Equations — with a particular emphasis on Beam Equations.

## Week 4

In Week 4 we will start looking in particular at simply supported beams.

## Assessment 1

Assessment 1 has been emailed to all of you. The hand in date in Thursday 12 October. Work handed in late shall be assigned a mark of zero.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

## Student Resources

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 2

We finished looking at algebra and ‘easy’ equations. We started looking then at quadratic equations.

## Week 3

In Week 3 we will finish looking at quadratic equations and look at exponents and logarithms.

## Assessment 1

I said last week that Assessment 1 will be on  Friday 13 October (Week 5). I am not so sure now: it might be pushed into Week 6. Expect a sample, an proper notice, at least a week beforehand.

## Study

Please feel free to ask me questions about the exercises via email or even better on this webpage — especially those of us who struggled in the test.

## Student Resources

A nice little question:

Given a regular pentagon with side length $s$what is the relationship between the area and the side-length?

First of all a pentagon:

We use triangulation to cut it into a number of triangles:

With $180^\circ$ in each of the three triangles, there $3\times 180^\circ=540^\circ$ in those angles around the edges, and, as there are five of them, they are each $108^\circ$.

Next triangulate from the centre. With a plain oul pentagon we might not be sure that such a centre exists but if you start with a circle and inscribe five equidistant points along the circle, the centre of the circle serves as this centre:

As everything is symmetric, each of these triangles are the same and the ‘rays’ are also the same as they are all radii. The angle at the centre is equal to $360^\circ/5=72^\circ$, and furthermore, by symmetry, the rays bisect the larger angles $108^\circ/2=54^\circ$ and so each of these triangles are $72^\circ,54^\circ,54^\circ$.

Using radians, because they are nicer, $\displaystyle 54^\circ=\frac{3\pi}{10}$. Note that, where $h$ is the perpendicular height:

$\displaystyle \tan\left(\frac{3\pi}{10}\right)=\frac{h}{s/2}\Rightarrow h=\frac{s}{2}\tan(3\pi/10)$.

A problem for another day is finding the exact value of $\tan\left(3\pi/10\right)$. It is

$\displaystyle \frac{\sqrt{5}+1}{\sqrt{10-2\sqrt{5}}}$

$\displaystyle \Rightarrow h=\frac{\sqrt{5}+1}{2\sqrt{10-2\sqrt{5}}}\cdot s.$

Therefore the area of one such triangle is:

$\displaystyle A(\Delta)=\frac{1}{2}s\cdot \frac{\sqrt{5}+1}{2\sqrt{10-2\sqrt{5}}}\cdot s=\frac{\sqrt{5}+1}{4\sqrt{10-2\sqrt{5}}}\cdot s^2$,

Therefore the area of the pentagon is five times this:

$\displaystyle A=5\cdot\frac{\sqrt{5}+1}{4\sqrt{10-2\sqrt{5}}}\cdot s^2$

$\displaystyle=\underbrace{\frac{5\sqrt{5}+5}{4\sqrt{10-2\sqrt{5}}}}_{=:\alpha}\cdot s^2$,

with $\alpha\approx 1.721$. It might be possible to simply $\alpha$ further.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 2

We continued working with the dot product and then introduced the cross product.

## Week 3

We will look at the applications of vectors to work and moments. We might begin Chapter 2: Matrices.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 2

We finished talking about linear least squares curve fitting. We spoke briefly about the Pearson correlation coefficient, and we began talking about fitting curves/models that aren’t of the form

$Y=a\,\theta_1(X)+b\,\theta_2(X)$

## Week 3

In Week 3 we will continue talking about these non-linear models.
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