How to get trig identities from the unit circle @THaWoM

The How and Why of Mathematics A lot of trigonometric identities don't need to be memorised - you can derive them on the fly using the unit circle. I'll be covering sin(180 + x) = -sin x, cos(-x) = cos x, etc, as well as cos^2 x + sin^2 x = 1 and equations you can derive from it, sin(90-x) = cos(x) etc, and the various ways of expressing cot. In this video we'll also start to use radians.

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

updated 5 years ago

How to get trig identities from the unit circle

The How and Why of Mathematics 2019-01-29 | A lot of trigonometric identities don't need to be memorised - you can derive them on the fly using the unit circle. I'll be covering sin(180 + x) = -sin x, cos(-x) = cos x, etc, as well as cos^2 x + sin^2 x = 1 and equations you can derive from it, sin(90-x) = cos(x) etc, and the various ways of expressing cot. In this video we'll also start to use radians.

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

To the families struggling with homeschooling/remote learning during lockdowns

The How and Why of Mathematics 2021-09-16 | I keep hearing about parents not having time to make sure their kids get schoolwork done in lockdown, so I want to tell you about the kind of homeschooling I did, which is a tried and tested method that requires minimal parental supervision. I'm talking about unschooling, which just means leaving a child in charge of their own education. I did unschooling for three years, then went on to complete a degree in physics with first class honours, getting the top marks at my uni in 1st and 2nd year maths, and 2nd and 3rd years physics.

This is a great article if you're looking for evidence that unschooling works: kqed.org/mindshift/37091/how-do-unschoolers-turn-out
Some excerpts:
"The respondents were overwhelmingly positive about their unschooling experience, saying it improved their children's general well-being as well as their learning, and also enhanced family harmony.
…
All but three of the 75 respondents felt the advantages of unschooling clearly outweighed the disadvantages. Almost all said they benefited from having had the time and freedom to discover and pursue their personal interests, giving them a head start on figuring out their career preferences and developing expertise in relevant areas. Seventy percent also said 'the experience enabled them to develop as highly self-motivated, self-directed individuals,'
…
Several themes emerged: Getting into college was typically a fairly smooth process for this group; they adjusted to the academics fairly easily, quickly picking up skills such as class note-taking or essay composition; and most felt at a distinct advantage due to their high self-motivation and capacity for self-direction.
…
In the words of one woman: 'I already had a wealth of experience with self-directed study. I knew how to motivate myself, manage my time, and complete assignments without the structure that most traditional students are accustomed to. … I know how to figure things out for myself and how to get help when I need it.'
…
All survey respondents were also asked about their employment status and career, and 63 answered a follow-up survey asking about their work in more detail. More than three-quarters of those who answered the follow-up survey said they were financially self-sufficient; the rest were either students, stay-at-home parents, or under the age of 21 and launching businesses while living at home.
…
The range of jobs and careers was very broad—from film production assistant to tall-ship bosun, urban planner, aerial wildlife photographer, and founder of a construction company—but a few generalizations emerged. Compared to the general population, an unusually high percentage of the survey respondents went on to careers in the creative arts—about half overall, rising to nearly four out of five in the always-unschooled group. Similarly, a high number of respondents (half of the men and about 20 percent of the women) went on to science, technology, engineering or math (STEM) careers.
…
He adds that this trend manifests across white- and blue-collar careers. 'In the Sudbury survey, there were people working as carpenters or auto mechanics, etcetera, but in situations where they were occupationally self-directed, set their own schedules, and solved their own problems, rather than shuffled papers, or worked on assembly lines where no original work was being done.' In other words, he says, unschoolers of all types had overwhelmingly chosen careers high in those qualities that sociologists have found lead to the highest levels of work satisfaction.
…
Parents' involvement levels with their children differed a lot, Gray says. Some were more hands-off, whereas others helped with learning, and in some cases even learned things (such as a foreign language) alongside their child, following the child's lead. 'All of those ways seem to work,' he says."

You might also be interested in a study of graduates from the Sudbury Valley School (SVS), as this is a very similar form of education (children not forced to attend lessons):
cdn2.psychologytoday.com/assets/attachments/1195/democratic-schooling-aje_0.pdf
Democratic Schooling: What Happens to Young People Who Have Charge of Their Own Education?
Author(s): Peter Gray and David Chanoff
Source: American Journal of Education, Vol. 94, No. 2, (Feb., 1986), pp. 182-213
Stable URL: http://www.jstor.org/stable/1084948

The books I mentioned at the end were "The Teenage Liberation Handbook: How to Quit School and Get a Real Life and Education" by Grace Llewellyn, and anything by John Holt (e.g. How Children Learn, Growing Without Schooling, Instead of Education).

A geometry construction problem I could actually do!

The How and Why of Mathematics 2020-12-29 | This is a geometry problem someone sent me on twitter. You really don't need anything more complicated than understanding isosceles triangles and angle sum in a triangle = 180°, but the solution does take some creativity with construction. In triangle ABC, AB=AC, and D is a point inside the triangle such that AD=BC, ∠BAD = 10°, and ∠CAD = 30°. Find ∠BDC.

I've added this to the AMC senior division playlist even though it's not from the AMC, because it's of a similar difficulty level.

Senior AMC 2019 Q30: function recursion

The How and Why of Mathematics 2020-09-10 | This question from the 2019 Australian Mathematics Competition senior paper involves a function with a recursive, piecewise definition.

How to do maths tutoring online

The How and Why of Mathematics 2020-04-13 | I've been doing maths tutoring online for a couple years now so I wanted to share how I do that with anyone wanting to switch over from face to face tutoring, due to the COVID-19 situation. This is also a video for businesses who make online whiteboards, to show them the kind of features that I, as a tutor, want from their software.

I talk about how you can use a stylus or webcam to show your student what you're writing; it can be a laptop with a built-in stylus, an external wacom tablet, a universal stylus with a touchscreen, or an external webcam pointed down at a notebook. I've found that it's not necessary for the student to also have a way of writing, as they can just talk me through what they would do, as I write what they're saying on the online whiteboard.

I forgot to talk about audio. I mostly just use a phone call, but sometimes skype or google hangouts.

Online whiteboards:
github.com/jennigorham/websketch - the whiteboard software I wrote and use
http://groupboard.com - a free online whiteboard
miro.com/online-whiteboard - another
zapier.com/blog/best-online-whiteboard - comparison of a bunch of different online whiteboards

Long division maths puzzle - fill in the boxes

The How and Why of Mathematics 2020-03-29 | Fun maths puzzle - fill in the digits to complete this long division problem. I love this question because it looks so impossible at first to work out what the digits have to be with so little information.

2019 NSW HSC Maths Ext2 Q16

The How and Why of Mathematics 2020-03-14 | Solutions for question 16 of the 2019 NSW HSC Mathematics Extension 2 exam, covering solving cubic equations, roots of quartics, trigonometry, complex numbers, sine rule, and circle geometry proofs.

Solutions for the 2018 exam: youtube.com/playlist?list=PLkNbGj38zTLNCI0H1sUiW0m3XAEUCdD4_
2017 exam: youtube.com/playlist?list=PLkNbGj38zTLMvmm3r6ZdZj9su_1VJ_TXB
2016 exam: youtube.com/playlist?list=PLkNbGj38zTLO9WeHfJ3LUzG_UvY5LQbf7

Maths Olympiad Question: IMO 2018 C1

The How and Why of Mathematics 2020-01-26 | My solution for the 2018 International Mathematical Olympiad first combinatorics question: "Let n ≥ 3 be an integer. Prove that there exists a set S of 2n positive integers satisfying the following property: For every m = 2, 3, . . . , n the set S can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality m."

I must apologise for my poor explanations, especially from 22:00 onward. I was going to remake that bit, but I'm apparently too lazy and freaked out by the bushfires to get anything done.

Bloopers and auto-captions gone wrong! Also merry Christmas!

The How and Why of Mathematics 2019-12-25 | A light, fun video for the holiday season. My stuff-ups and YouTube's automatic captioning failures. No pigeons were harmed in the making of this video.

Hardest Canadian maths questions - Mathematics 30-1 Diploma Exams

The How and Why of Mathematics 2019-12-02 | I go over the trickiest questions from the Mathematics 30-1 released items (from Alberta, Canada), covering trigonometry, combinatorics, functions and their graphs, exponents and logarithms.

Other videos referenced:
Trig identities: youtube.com/watch?v=Yx77uXi7UIk
Solving trig equations with exact solutions: youtube.com/watch?v=-ldZSvglx1A
Solving trig equations with a calculator: youtube.com/watch?v=igkTyHWXx7Y
Expanding (a+b)³: youtube.com/watch?v=kBjd0RBC6kQ
Talent quest combinatorics: youtube.com/watch?v=jJh4-crV6qg

Questions covered:
MC7 from http://ncee.org/wp-content/uploads/2017/01/Alb-non-AV-11-Diploma-Exams-Released-Materials-Math-2014.pdf
MC7, MC15, MC25 from education.alberta.ca/media/3653447/03-math-30-1-releaseditems-2017_20170830.pdf
NR6, MC6, MC9, MC18, MC22 from education.alberta.ca/media/3272898/02-math-30-1-released-items-2016-17.pdf

Akshats questions, and what Ive been up to

The How and Why of Mathematics 2019-11-17 | Sorry for not uploading a video for ages! A few months ago, one of my subscribers, Akshat, asked me these geometry questions, and I haven't been able to answer them, so I'm opening them up to the public so hopefully someone can post a solution. Also an update on what's kept me busy lately - house hunting, and getting into making music (learning how to use LMMS and writing a software synthesizer).

Hardest Maths Questions (Junior) - Candles

The How and Why of Mathematics 2019-08-30 | Only 5% of the junior students got this Australian Mathematics Competition question right (8% of the intermediate students, and 15% of the senior students got it right). There are two candles: a shorter one that burns for 10 hours and a longer one which burns for 7 hours. After 4 hours of burning, the candles are the same height. What is the ratio of the shorter candle's height divided by the longer candles height? To answer the question I'll show you how to write down the information as an algebraic equation. Many people struggle with this (turning a word problem into an equation), so I'll show you how to do a guess and check first and then turn that into an equation.

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

See the full junior playlist here: youtube.com/watch?v=QDnT4evo-5w&list=PLkNbGj38zTLMfJ7ThxPWKAzLQP6XrD1zz&index=2
I also have videos on senior AMC questions: youtube.com/watch?v=JoVB72-KSk4&list=PLkNbGj38zTLOO9aODJVznsquLhi1WeyFo&index=2

The Problem with PEMDAS: Why Calculators Disagree

The How and Why of Mathematics 2019-08-05 | Some calculators say 6/2(1+2) = 1 and others say it equals 9 (similarly 8 divided by 2(2+2) can be 1 or 16 depending on the calculator). How did this disagreement on the order of operations come to be? My first PEMDAS video focused on how mathematicians, scientists and engineers interpret expressions; this video focuses on how calculators treat them. It turns out that the rule that juxtaposition comes before division is much older than "PEMDAS", and has been widely used for decades. So why did some calculator brands switch from this rule (which I call "PEJMDAS") to treating juxtaposition as the same priority level as division? And what can we do about the ambiguity?

See my first PEMDAS video here: youtu.be/lLCDca6dYpA

See also David Linkletter's article: plus.maths.org/content/pemdas-paradox

References:
Sass' article: http://www.math.ucdenver.edu/~jloats/Student%20pdfs/4_Order%20of%20OperationsSass.pdf
First year algebra: archive.org/details/firstyearalgebra00well/page/18 also p85
First course in algebra: archive.org/details/firstcourseinal01toutgoog/page/n23 (p10) also p74 (page 90 of the pdf)
Second course in algebra: archive.org/details/secondcourseinal00wellrich/page/4 also p64
Lennes' article, 'Relating to the Order of Operations in Algebra': jstor.org/stable/2972726

Sharp EL-512 manual: manualslib.com/manual/1177727/Sharp-El-512.html?page=9#manual (p14)
TI 81 manual: manualslib.com/manual/325929/Texas-Instruments-Ti-81.html?page=34#manual (p1-8)
TI website's comment on the issue: epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110

AMS Guide for Reviewers May 2000 web.archive.org/web/20000815202937/http://www.ams.org/authors/guide-reviewers.html
APS Physical Review Style and Notation Guide cdn.journals.aps.org/files/styleguide-pr.pdf p21
AIP style guide: http://web.mit.edu/me-ugoffice/communication/aip_style_4thed.pdf p23 (page 26 of the pdf)

Maths Olympiad Questions - 2019 INMO Q1

The How and Why of Mathematics 2019-07-19 | A circle geometry problem from the 2019 Indian National Mathematical Olympiad. I'll go through two solutions, the first an elegant classical/synthetic proof, and the second a messy (but straightforward to come up with) modern solution using the sine rule and lots of algebra.

The elegant solution comes from artofproblemsolving.com/community/c6h1770750p11619132

How computers can help with maths

The How and Why of Mathematics 2019-07-05 | One of my viewers asked me the following question: "Let ABCD be an isosceles trapezoid with (AB parallel to CD). Let E be the mid-point of AC. Denote by omega and ohm the circumcircles of triangles ABE and CDE, respectively. Let P be the crossing point of the tangent to omega at A with the tangent to ohm at D. Prove that PE is tangent to ohm."
I couldn't think of a proof using classical (synthetic) geometry, so decided to use analytic geometry, and the algebra got a bit messy so I used a computer algebra system (SageMath) to help out. So this video is just a little intro to using computers to help out with maths. I also go off on a tangent about problem-solving techniques and how to come up with a strategy for geometry proofs.

You can try out SageMath (without installing) at: http://sagecell.sagemath.org
Or install SageMath: http://www.sagemath.org/download.html
For linux, it says "At the moment there are no distribution-specific packages available. Progress is being made for Debian and Fedora." but I think that's out of date because I installed it on Ubuntu with "sudo apt install sagemath".
SageMath tutorial: doc.sagemath.org/html/en/tutorial

MetaPost tutorials: tug.org/metapost.html
Installing metapost is usually done by installing TeX Live tug.org/texlive or MikTeX miktex.org

MetaPost source code for the diagram: github.com/jennigorham/metapost-examples/blob/master/isosceles-trapezoid.mp

JEE advanced maths 2018 - the parts I struggled with

The How and Why of Mathematics 2019-05-21 | Questions 4,5, and 6 from paper 1 and question 7 from paper 2 were the ones that I got stuck on from the 2018 JEE Advanced maths sections. Now that I've figured them out, I'll go through the solutions in this video.

Wikipedia pages mentioned in this video:
Mean value theorem: en.wikipedia.org/wiki/Mean_value_theorem
Examples of differential equations: en.wikipedia.org/wiki/Examples_of_differential_equations
Leibniz integral rule: en.wikipedia.org/wiki/Leibniz_integral_rule
Method of undetermined coefficients: en.wikipedia.org/wiki/Method_of_undetermined_coefficients
Integrating factor method: en.wikipedia.org/wiki/Integrating_factor
Monotonic functions: en.wikipedia.org/wiki/Monotonic_function

JEE advanced - probability/combinatorics questions

The How and Why of Mathematics 2019-05-07 | Questions 17 and 18 from the 2018 JEE Advanced Paper 1 cover probability and combinatorics. Q17 is about derangements, but I won't be using a formula for that; I'll just show you how to use systematic listing to find the answer. I'll also be covering what set intersection and union mean in the context of events in probability.

Past JEE advanced papers can be downloaded from https://jeeadv.ac.in/archives/past-question-papers.html

Maths Olympiad Questions - 2017 IMO C1

The How and Why of Mathematics 2019-04-23 | I'm starting a series of videos on olympiad questions from the International Mathematical Olympiad, the Australian Mathematical Olympiad, the Putnam competition, and any others that take my fancy. This one covers question C1 (C for combinatorics) from the 2017 IMO.

The olympiad paper can be found at http://imo-official.org/problems/IMO2017SL.pdf
The Martin Gardner book I mentioned is "My Best Mathematical and Logic Puzzles" amazon.com/Mathematical-Logic-Puzzles-Dover-Recreational-ebook/dp/B00DP7UJMY
The other author I mentioned is Ian Stewart: amazon.com/Ian-Stewart/e/B00URW7B2G
Paul Lockhart's article, "A Mathematician's Lament" can be found at maa.org/external_archive/devlin/LockhartsLament.pdf

Hardest maths questions - minimisation

The How and Why of Mathematics 2019-04-09 | This question, from the 2001 senior Australian Mathematics Competition, requires us to find the minimum value of a complicated algebraic expression. Let's see if we can find a geometric interpretation of the expression, to avoid using calculus (but watch from 6:46 to see the solution using calculus).

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

Making of: refactorisation, solids of integration, and computer graphics in future videos

The How and Why of Mathematics 2019-04-01 | Just a chat about programming in general, and how I made the refactorisation video, the 3D models for solids of integration in past exam papers, and my plans for computer graphics in future videos.

Python tutorials: wiki.python.org/moin/BeginnersGuide/NonProgrammers
Download Ubuntu (linux distribution): ubuntu.com/download/desktop
Note: if you want to run linux and mac/windows on the same computer, then google "dual-boot ubuntu mac" or "dual-boot ubuntu windows" to find a guide.
Project Euler: projecteuler.net
My FreeCAD python code: github.com/jennigorham/Volume-by-integration-in-freecad

General solutions and how to solve sin(4x+60°) √3/2 etc

The How and Why of Mathematics 2019-03-19 | When solving equations such as cos(3x-30°) = 1/2, it helps to know the general solution. I'll go through the general solutions to trig equations involving sine, cos, and tan, in both degrees and radians. I'll teach you some tricks to make these easy, using the unit circle, and a calculator.

Hardest maths questions - refactorisation and prime numbers

The How and Why of Mathematics 2019-03-12 | Express 8×9×10×11×12×13×14 as another product of consecutive whole numbers. How can we factorise a number without calculating it? This question from the AMC junior division was answered correctly by 0% of students! To solve it, I'll show you how composite numbers can be pulled apart into their prime factors and recombined in new ways, which is a trick I use in a lot of different situations (mental arithmetic, finding an LCM when adding fractions or doing elimination, factorising quadratics, etc.).

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

See the full junior playlist here: youtube.com/watch?v=QDnT4evo-5w&list=PLkNbGj38zTLMfJ7ThxPWKAzLQP6XrD1zz&index=2
I also have videos on senior AMC questions: youtube.com/watch?v=JoVB72-KSk4&list=PLkNbGj38zTLOO9aODJVznsquLhi1WeyFo&index=2

Hardest maths questions: simultaneous diophantine equations

The How and Why of Mathematics 2019-03-04 | This question from the 2007 Australian Mathematics Competition (senior division) involves 3 simultaneous equations with positive integer solutions. We have to find the smallest value that x can take. 0% of students got this one right! I'll be making heavy use of the elimination method in this video, although you could use substitution instead.

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

How to solve trig equations with a calculator (and the unit circle)

The How and Why of Mathematics 2019-02-26 | Learn how to use a calculator (Casio, TI, HP, etc) to solve equations such as sin(theta) = 0.2, cos(theta) = -0.6, tan(theta) = -2, etc, finding all solutions in a given range, for degrees or radians. You do this by using the inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹). I'll also show you how to find solutions to the nearest minute (how to convert to degrees, minutes, and seconds on a calculator). What's also important is being able to switch your calculator to degrees or radians mode.

Solve trig equations with exact solutions - the easy way (unit circle)

The How and Why of Mathematics 2019-02-18 | A simple way to visualise trigonometric equations lets us solve them quickly using spatial intuition. No need for CAST or ASTC (all stations to central/all students take calculus) mnemonics. I remember Dr Hoffman saying "This is how trigonometry should be taught in schools" before he showed us these techniques.

In this video I'll just be covering simple trig equations with exact solutions, for both degrees and radians. For example, equations like sin theta = -1/2, cos theta = sqrt(2)/2, tan theta = -sqrt(3), sin theta = -sqrt(3)/2, cos theta = 1/2, tan theta = -1, and sin theta = 0. And I'll show you how to work with radians so you can find angles like 11pi/6 or 5pi/4 on the unit circle.

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

Upcoming videos, thank you to my first 100 subscribers, and happy birthday to me!

The How and Why of Mathematics 2019-02-12 | I've just reached 100 subscribers!! What a perfect birthday present. I wanted to thank everyone who's subscribed so far, and tell you about some upcoming videos I'm planning to make in 2019. Let me know what you would like me to focus on.

Special angles: exact values of sin, cos, and tan, using the unit circle

The How and Why of Mathematics 2019-02-05 | The unit circle is an underused tool for figuring out which exact values go with which special angle (30, 45, 60, 90, 120, 135, ... or pi/6, pi/4, pi/3, pi/2, 2pi/3, 3pi/4, ...). Using the definitions of cos as the x-coordinate, sine as the y-coordinate, and tan as the slope, this becomes much easier since the values of sin, cos, and tan can be estimated visually. As usual, understanding is better than brute force memorisation.

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

The unit circle - secret tricks to make trigonometry easy

The How and Why of Mathematics 2019-01-22 | Solving trig equations, finding sin, cos, and tan of special angles, and certain trig identities are all much easier with these tricks, which I learned from Dr Norm Hoffman in afterschool mathematical problem-solving classes, and I haven't seen these techniques taught anywhere else.

In this introductory video I'll cover the definitions of sine, cosine, and tangent in the unit circle, and explain how angles are drawn in trigonometry (anticlockwise from the positive x-axis). Once you get the hang of cosine = x-coordinate, sine = y-coordinate, and tangent = slope, trigonometry becomes much easier.

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

Hardest maths questions - tyres

The How and Why of Mathematics 2019-01-18 | How far can you drive without wearing down your tyres too much? This question comes from the 1990 Australian Mathematics Competition, and only 8% of the junior students got it right. This is another great question for people learning algebra, because you can practise turning a word problem into an equation.

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

The car image in the thumbnail is by Ömer Uzun (deviantart.com/qmer/art/Blue-Car-123991100) under creative commons (creativecommons.org/licenses/by/3.0/).

Hardest maths questions - find a^4 + b^4 + c^4

The How and Why of Mathematics 2019-01-15 | This was a question from the 2006 Australian Mathematics Competition senior division (for students around 17 years old) and only 1% got it right. Simultaneous equations - do we use substitution method, elimination method, or just throw everything at the wall and see what sticks?

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

Hardest maths questions - inheritance

The How and Why of Mathematics 2019-01-11 | How do you turn a word problem into an equation to solve? This question comes from the 1983 Australian Mathematics Competition, and only 5% of the junior students got it right. This is a great question for people learning algebra, because it'll give you an overview of what algebra is about, and an appreciation for how useful algebra is! You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

Hardest maths questions - solve equation involving cube roots

The How and Why of Mathematics 2019-01-08 | This was a question from the 1988 Australian Mathematics Competition senior division (for students around 17 years old) and only 3% got it right. It's a question about solving an equation involving cube and square roots, and is surprisingly linked with the history of complex numbers!

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

Hardest maths questions - functions and recursion

The How and Why of Mathematics 2018-12-22 | This was a question from the Australian Mathematics Competition senior division (for students around 17 years old) and only 3% got it right. It's a question involving recursion and function notation.

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

2018 NSW HSC Mathematics Extension 2 exam Q16

The How and Why of Mathematics 2018-12-18 | Solutions for question 16 from the 2018 NSW HSC Mathematics Extension 2 exam. This question covers induction, geometric proofs using similar triangles, roots of cubic equations and complex numbers.

2018 NSW HSC Mathematics Extension 2 exam Q15

The How and Why of Mathematics 2018-12-14 | Solutions for question 15 from the 2018 NSW HSC Mathematics Extension 2 exam. This question covers ellipses (conic sections), trig identities, De Moivre's theorem, and proving inequalities.

2018 NSW HSC Mathematics Extension 2 exam Q14

The How and Why of Mathematics 2018-12-11 | Solutions for question 14 from the 2018 NSW HSC Mathematics Extension 2 exam. This question covers integration by t-substitution, solving for speed when we know a in terms of v, integration by parts, and probability.

2018 NSW HSC Mathematics Extension 2 exam Q13

The How and Why of Mathematics 2018-12-07 | Solutions for question 13 from the 2018 NSW HSC Mathematics Extension 2 exam. This question covers solids of integration (method of cylindrical shells), finding the modulus and argument of a complex number, mechanics (horizontal and vertical forces with circular motion), hyperbolae and parabolae.

See github.com/jennigorham/Volume-by-integration-in-freecad for the 3D models.

2018 NSW HSC Mathematics Extension 2 exam Q12

The How and Why of Mathematics 2018-12-05 | Solutions for question 12 from the 2018 NSW HSC Mathematics Extension 2 exam. This question covers solids of integration, implicit differentiation, integrating a rational function with arctan, and graphing.

See github.com/jennigorham/Volume-by-integration-in-freecad for the 3D models.

2018 NSW HSC Mathematics Extension 2 exam Q11

The How and Why of Mathematics 2018-12-01 | Solutions for question 11 from the 2018 NSW HSC Mathematics Extension 2 exam. This question covers complex numbers, cubics with double roots, partial fractions, and a geometric proof (circle geometry). I'll also show you how to use the Casio fx-100AU calculator with complex numbers.

2018 NSW HSC Mathematics Extension 2 exam Q1-10

The How and Why of Mathematics 2018-11-27 | Solutions for questions 1 through 10 from the 2018 NSW HSC Mathematics Extension 2 exam. These cover integration, hyperbola asymptotes, coefficients of cubic equations, graphing, solids of revolution, complex numbers, graphing a function given the graph of its slope, and more.

See github.com/jennigorham/Volume-by-integration-in-freecad for the 3D models.

Perpendicular slope equation - a very simple explanation

The How and Why of Mathematics 2018-11-20 | Ever wonder where the negative reciprocal formula comes from? (i.e. m1 × m2 = -1.) See a very intuitive reason why, and learn how to work out the equation of a line that's perpendicular to another line with known slope, even if the original line is in general form.

Polynomial long division with complex coefficients examples

The How and Why of Mathematics 2018-11-16 | Combine polynomial long division with complex numbers for an extra challenge! I go over two examples in this video, showing you how to multiply and subtract the complex coefficients.

Hardest math questions - talent quest combinatorics

The How and Why of Mathematics 2018-11-04 | In how many ways can three judges at a talent quest rank the three performers such that two judges agree and the other one disagrees? Find out in this video, as we explore combinatorics, factorials, and systematic counting. This question comes from the 1981 Australian Mathematics Competition. You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

Maths exam walkthrough/solutions: 2016 HSC Mathematics Extension 2 missing parts

The How and Why of Mathematics 2018-10-24 | Sorry, I accidentally left these parts off for some reason (question 16c parts iv and v, which was about derangements and mathematical induction).

How to integrate fractions - lots of partial fractions examples

The How and Why of Mathematics 2018-10-11 | Integrating something like (3x + 12)/(x^2 + x - 2) (a fraction of a polynomial over another polynomial) requires partial fraction decomposition to split it up. In this video I go over several examples of how to use partial fractions, including some from past exam papers (the NSW HSC extension 2 maths exams). In each case I show you two different methods: equating the coefficients, and picking particular values of x.

Hardest maths questions - 2018 AMC senior division

The How and Why of Mathematics 2018-09-25 | Solutions for some of the hardest questions from the 2018 Australian Mathematics Competition senior division (for year 11 or year 12 students). There were a few questions here that were very geometry-focused, involving similar triangles or circle geometry,

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources

See the full playlist of senior questions at: youtube.com/playlist?list=PLkNbGj38zTLOO9aODJVznsquLhi1WeyFo
The video on the hardest junior questions is here: youtu.be/QDnT4evo-5w

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

Hardest math questions: |x| + |y| + |x+y| ≤ 2, area = ?

The How and Why of Mathematics 2018-09-24 | In this series of videos I'll be going over some of the hardest maths questions from the senior division (for year 11 & 12) of the past Australian Mathematics Competition papers. If you like tricky riddles and puzzles then you'll love these. Today's challenge is to find the area of the region defined by the inequality |x| + |y| + |x+y| ≤ 2. I'll walk you through dealing with the absolute values and graphing the region.

You can buy past AMC exam booklets at https://shop.amt.edu.au/collections/amc-resources
The video on the hardest junior questions is here: youtu.be/QDnT4evo-5w

I'm an online maths tutor. Go to http://thawom.com/tutoring to learn more.
Twitter: twitter.com/thawom0

How to find the y-intercept and equation of a line from coordinates

The How and Why of Mathematics 2018-09-16 | How can we work out the y-intercept and equation of a line from the coordinates of two points on the line, or from the slope and coordinates of one point? You'll need to be creative to work it out, because mathematics is about puzzles, not about following instructions. There are at least three ways you can figure out the y-intercept (and the equation in slope intercept form) from what you already know.

Go to http://thawom.com/practice/import.php?import=y_intercept_from_slope_and_point for the practice questions.

I'm an online maths tutor. See http://thawom.com/tutoring to find out more.
You can also follow me on twitter: twitter.com/thawom0

How to find gradient or slope given two points on a straight line

The How and Why of Mathematics 2018-09-11 | Work out the formula for slope/gradient with me! I'll give you lots of examples of how to calculate slope from the coordinates of two points on a straight line.

The practice questions can be found at: http://thawom.com/practice/import.php?import=slope_two_points

I'm an online maths tutor. Go to http://thawom.com/tutoring to find out more.
You can also follow me on twitter: twitter.com/thawom0

NSW HSC 2017 Maths Extension 2 Exam Solutions Q16 part7/7

The How and Why of Mathematics 2018-07-27 | Solutions for the NSW HSC 2017 Mathematics Extension 2 exam. This video covers question 16, on complex numbers, conic sections, mathematical induction, and combinatorics. Go to youtube.com/playlist?list=PLkNbGj38zTLMvmm3r6ZdZj9su_1VJ_TXB to see the other questions in this exam, or check out youtube.com/playlist?list=PLkNbGj38zTLO9WeHfJ3LUzG_UvY5LQbf7 for the 2016 exam.

I'm an online maths tutor. Go to http://thawom.com/tutoring to find out more.
Twitter: twitter.com/thawom0