# Singapore Maths Tuition

Mathematics Tuition Singapore

## Friday, 8 January 2016

## Thursday, 10 December 2015

## Tuesday, 24 March 2015

### Calculate UAS Points

# Calculate UAS Points / JC Ranking Points

# How to Calculate JC Ranking Points?

Check out this JC Ranking Points site to double check your calculations and to learn how the JC Ranking Points for University admission is calculated!Site: http://mathtuition88.com/2014/03/10/calculate-ranking-points-jc/

# Math Notes bundled with Free Exam Papers

Also check out our Highly Condensed Math Notes bundled with Free Exam Papers for practice! E Maths / A Maths / H2 Maths Notes available.Site: http://mathtuition88.com/math-notes-worksheets-sale/

## Sunday, 15 February 2015

### How to get First Class Honours or Summa Cum Laude

Recently I graduated with First Class Honours (Mathematics) and I would like to share some tips on how to achieve it!

Disclaimer: This post is not to boast about my achievements, but rather to teach other university undergraduates some strategies to achieve their goal. Getting First Class Honours (or Summa Cum Laude) is merely a humble and small achievement to be honest, but for many students it would be a very helpful thing to have for their future career or further studies. Hence, I sincerely hope this post will give some idea how to achieve the goal of getting First Class Honors. Students aiming for the very respectable grade classification of Second Upper Honours (or magna cum laude) would also definitely find this post useful.

Students who wish to read more about how to find your ideal college or university may want to read this excellent book The College Solution: A Guide for Everyone Looking for the Right School at the Right Price (2nd Edition). Also useful is 1001 Things Every College Student Needs to Know: (Like Buying Your Books Before Exams Start).

There is a joke that most university students only have time to choose two out of the following: Good Grades, Social Life, and Enough Sleep.

There is a time management life hack that I personally use and find it very useful. It is called the Pomodoro Technique. This technique can allow the busy college students to achieve some studying in bursts of 30 minutes, which is proven to be the optimal attention span of an average human! The Pomodoro Method is basically about studying / working for 25 minutes, and then taking a short break of around 5 minutes. Every four cycles, take a longer break (e.g. 15-30 minutes). This allows students to break a huge task into smaller and more manageable chunks of 30 minutes. Pomodoro Technique Illustrated (Pragmatic Life) has more details on this amazing time management technique.

Another important time management technique is what to do during long commutes on the train or bus. I live in one of the smallest countries in the world (Singapore), but my daily commute to the university could be as long as 2 hours (to and fro). Reading some notes or books during the traveling time could be really productive over the long term. If you prefer reading ebooks, you could consider the Amazon Kindle Paperwhite with Special Offers, Wi-Fi, Black which is one of the best ebook readers out there.

Another tip for those suffering from sinus or nose problems is to try nasal irrigation or sinus rinse. It took me almost 10 years to discover this. I was suffering badly from sneezing and sinus problems almost monthly. After using this NeilMed Sinus Rinse - 2 Bottles - 250 Premixed Packets I am 95% free from my previous sinus issues which were quite bad. I have been recommending this to almost everyone that I know who has sinus issues.

If your university has an option for an overseas exchange, do check it out. Sometimes modules taken on an overseas exchange are do not count for your GPA, and hence has an effect of preserving your grades. For example, if your GPA is currently enough to get a summa cum laude provided you maintain your score, going on an overseas exchange programme would freeze your GPA or CAP. Also, overseas exchange modules are notoriously easy to pass. Hence, if you pass all your exchange programme modules, you have effectively maintained your summa cum laude or First Class Honours grade at zero risk.

Another strategy is to go for easier modules to increase your GPA or CAP, effectively pulling them up. By now you would know that not all courses are graded equally, and some courses require much less effort but hand out As more liberally. However, please take note that this should not be the main objective of your studies at college, since taking courses like the proverbial "underwater basket weaving" courses do not improve your knowledge in any way! Hence, this technique should be used only sparingly.

Disclaimer: This post is not to boast about my achievements, but rather to teach other university undergraduates some strategies to achieve their goal. Getting First Class Honours (or Summa Cum Laude) is merely a humble and small achievement to be honest, but for many students it would be a very helpful thing to have for their future career or further studies. Hence, I sincerely hope this post will give some idea how to achieve the goal of getting First Class Honors. Students aiming for the very respectable grade classification of Second Upper Honours (or magna cum laude) would also definitely find this post useful.

Students who wish to read more about how to find your ideal college or university may want to read this excellent book The College Solution: A Guide for Everyone Looking for the Right School at the Right Price (2nd Edition). Also useful is 1001 Things Every College Student Needs to Know: (Like Buying Your Books Before Exams Start).

### What is First Class Honours?

Majority of students at the undergraduate level would have heard of the term 1st Class Honours, but some may be unsure of what it means. First Class Honours is a terminology traditionally used in British universities or universities based on the UK system. It represents one of the highest achievements attainable in terms of academics for undergraduates. Usually students would have to achieve a certain GPA (Grade Point Average) and also satisfy certain additional requirements such as getting a certain grade for the final honours thesis. The term Summa Cum Laude (latin for "with highest honors") is usually used in American universities instead, but it is basically the same thing. For the university I graduated from, the National University of Singapore (NUS), students need to get a CAP (Cumulative Average Point) of at least 4.5/5, in addition to at least an A- for the final year project (this requirement has since been scrapped).### Is it Difficult to get Summa Cum Laude

Most schools will try to prevent grade inflation by controlling the number of students who get First Class Honours. This may be done through bell curving grades for individual modules. Hence, the percentage of students who get Summa Cum Laude may be kept low, though this percentage can vary from university to university. Hence, getting First Class Honours takes planning and strategy as it may be challenging to attain.### How to get Summa Cum Laude/Magna Cum Laude

Congratulations for finding this website, as I will endeavor to share with you all the tips for getting the First Class Honours or Summa Cum Laude. Ideally, the earlier you start planning and making your strategy, the better. Since Summa Cum Laude depends on your GPA (Grade Point Average), every year matters. Many students start planning too late, and even though their grade improves their earlier GPA drags their overall grades down, making getting the Summa Cum Laude difficult or even impossible.### Study Consistently and Revise Constantly

This first tip may seem like a cliche or no-brainer, but essentially it is the key to achieving your Summa Cum Laude. Never leave your revision to the last minute. Do tutorials on time even though you may not be required to hand in them. Clarify doubts as soon as possible to prevent them from piling up. Check out this book Study Strategies Made Easy: A Practical Plan for School Success for some study strategies that are both practical and concrete.### Time Management

For most people, time is really tight in college. 24 hours a day is barely enough to all the things that a college major is supposed to do.There is a joke that most university students only have time to choose two out of the following: Good Grades, Social Life, and Enough Sleep.

There is a time management life hack that I personally use and find it very useful. It is called the Pomodoro Technique. This technique can allow the busy college students to achieve some studying in bursts of 30 minutes, which is proven to be the optimal attention span of an average human! The Pomodoro Method is basically about studying / working for 25 minutes, and then taking a short break of around 5 minutes. Every four cycles, take a longer break (e.g. 15-30 minutes). This allows students to break a huge task into smaller and more manageable chunks of 30 minutes. Pomodoro Technique Illustrated (Pragmatic Life) has more details on this amazing time management technique.

Another important time management technique is what to do during long commutes on the train or bus. I live in one of the smallest countries in the world (Singapore), but my daily commute to the university could be as long as 2 hours (to and fro). Reading some notes or books during the traveling time could be really productive over the long term. If you prefer reading ebooks, you could consider the Amazon Kindle Paperwhite with Special Offers, Wi-Fi, Black which is one of the best ebook readers out there.

### Get Enough Sleep

Getting enough sleep is essential if you are aiming for a First Class Honours or Summa Cum Laude. Trying to cut down on sleep may lead to sleep deprivation which is not good. Sleep deprivation may lead to depression, weight gain, or even worse things like heart diseases. Most people need around 7 to 8 hours of sleep to function optimally. There are some people out there with special genes that make them need less sleep. Margaret Thatcher was one of the famous people who only needed 4 hours of sleep a night! Unfortunately, this gene is rare and most people don't have it. Personally, I am a long sleeper who needs 8 to 9 hours of sleep to function optimally. This is a major disadvantage, to be honest, when comparing with those people who can survive on 6 hours of sleep (or less!) However, I still managed to get Summa Cum Laude, and you can do it too! Remember, it is not worth it to sacrifice your health to get Summa Cum Laude or First Class Honours... One trick is to take power naps. Power naps as short as 5-8 minutes have been proven to be effective to raise alertness! I took power naps in the school library (the most comfortable place in school to sleep). It was a bit embarrassing at first, but many other people were also doing the same. :) I read this book Take a Nap! Change Your Life. and it gave me plenty of ideas of how to fit in power naps to get enough sleep, and also some interesting info on polyphasic sleeping and sleep cycles.### Health Issues

Health issues is incredibly important if you are aiming to get Summa Cum Laude or First Class Honours. First, mental health is very important to the student. Signs of depression and insomnia must be taken seriously and help be sought as soon as possible. Nutrition is also important, the brain consumes a massive amount of energy and we need to supply the brain cells with enough nutrition and vitamins. Personally, I drink this BRAND'S Essence of Chicken (6 x 2.3 fl oz.) which is a good alternative to caffeine. For me, coffee is only useful as a last resort, since I experience a caffeine slump after the effects of caffeine wear off.Another tip for those suffering from sinus or nose problems is to try nasal irrigation or sinus rinse. It took me almost 10 years to discover this. I was suffering badly from sneezing and sinus problems almost monthly. After using this NeilMed Sinus Rinse - 2 Bottles - 250 Premixed Packets I am 95% free from my previous sinus issues which were quite bad. I have been recommending this to almost everyone that I know who has sinus issues.

### Subject Specific Advice

Personally, I majored in Mathematics, and here I would like to give some advice to undergraduates majoring in math. The transition from high school math to undergraduate math is a big one. The most important difference being the idea of proofs. Undergraduate math is mostly about proving, and that is usually the hardest part. Also, finding a good Math textbook is incredibly important. I have listed a few of the very best Math textbooks at my website here.Also, while reading those Math books, be sure to try out some of the exercises as Math is all about practice rather than rote learning.### Seek help from Friends and Seniors

This is also a good tip that can be really helpful. Borrowing notes or previous test exams from friends and seniors who took the course a few semesters earlier can literally improve your grade by a huge amount. I had a good friend who lent me some notes and previous quizzes which he took the previous semester. Working through the test papers last semester was a major advantage for me, as I realized during the exam that the lecturer recycled some questions from the previous semester! The question, though slightly modified and different, were similar in style and format. Most students will agree that this is a huge and significant advantage, and may be the key to achieving your summa cum laude or first class honours. This technique really works because most of the time, professors are really busy people who don't really have time to set novel and new questions each semester. They will most often modify existing questions from the previous semester to generate new questions for the current semester.### Gaming the GPA System (Legally)

This section is about how to game the university or college system to achieve your GPA. Of course, this needs to be done legally, and depends on each university or college.If your university has an option for an overseas exchange, do check it out. Sometimes modules taken on an overseas exchange are do not count for your GPA, and hence has an effect of preserving your grades. For example, if your GPA is currently enough to get a summa cum laude provided you maintain your score, going on an overseas exchange programme would freeze your GPA or CAP. Also, overseas exchange modules are notoriously easy to pass. Hence, if you pass all your exchange programme modules, you have effectively maintained your summa cum laude or First Class Honours grade at zero risk.

Another strategy is to go for easier modules to increase your GPA or CAP, effectively pulling them up. By now you would know that not all courses are graded equally, and some courses require much less effort but hand out As more liberally. However, please take note that this should not be the main objective of your studies at college, since taking courses like the proverbial "underwater basket weaving" courses do not improve your knowledge in any way! Hence, this technique should be used only sparingly.

### The End

That's about all I have to share, hope it was helpful! Do post any comments below, especially if you have any other tips to share. Thanks and good luck!### The Math of Valentine's Day and Love

Valentine's Day is just over. Maths is really useful, watch this humorous video to learn how Math can help you find your true love!

## Sunday, 8 February 2015

### How to Flip a Coin to Land Heads Up

One good book by Professor Persi Diaconis is: Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks. A professional mathematician and also a magician, Persi Diaconis reveals the secrets of amazing, fun-to-perform card tricks--and the profound mathematical ideas behind them--that will astound even the most accomplished magician.

## Saturday, 7 February 2015

### A Math comic that only Math majors will understand

This is really a comic that Math Majors will empathize with...

It is probably due to lazy textbook authors who leave almost everything as an exercise, and every other theorem is "trivial". Such a textbook is of very little pedagogical value.

On the other hand, an example of a good Math Textbook is Abstract Algebra, 3rd Edition by Dummit and Foote. It is one of the most widely acclaimed textbooks for Abstract Algebra, one of the most arcane and esoteric fields of Math. I have also written a book review on Topology by James R. Munkres, another excellent Math book.

## Thursday, 5 February 2015

### RI/RJC Free Prelim Math Full Solution

Let's go through how to solve this RI/RJC Prelim Vectors Question.

(i)

First, we can note that we can let $\mu=0$ and $\lambda=1$ for convenience since the vector equation holds for all values of $\lambda , \mu$. Then we have $\textbf{r}=\left( \begin{array}{c}6\\t\\2\end{array}\right)$. Putting this into the given equation in (i), we have $12+3t+2=5$, hence $t=-3$.

(ii) Let M be the midpoint of AB. Note that M will lie on the plane $\pi_1$! By the Midpoint Theorem,

$\vec{OM}=\frac{1}{2}[\vec{OA}+\vec{OB}]=\left( \begin{array}{c}4\\-6\\7+\frac{1}{2}s\end{array}\right)$.

Since M lies on the plane, $\vec{OM}\cdot \left(\begin{array}{c}2\\3\\1\end{array}\right)=5$. Expanding the dot product, we get $s=16$.

(iii) We need to show that the normals of the two planes are perpendicular.

$\mathbf{n_1}=\left( \begin{array}{c}2\\3\\1\end{array}\right)$ First we note that B also lies on $\pi_2$. We can obtain a normal of $\pi_2$ by taking the cross product $\mathbf{n_2}=\vec{AB}\times\left( \begin{array}{c}3\\-2\\0\end{array}\right)=\left( \begin{array}{c}4\\6\\-26\end{array}\right)$

We then calculate that $\mathbf{n_1}\cdot\mathbf{n_2}=0$. (shown)

If we draw a diagram, we can see that the shortest distance from A to the line of intersection is actually $|\vec{AM}|$, where M is the midpoint found earlier. $\vec{AM}=\vec{OM}-\vec{OA}=\left(\begin{array}{c}2\\3\\1\end{array}\right)$. Thus $|\vec{AM}|=\sqrt{2^2+3^2+1^2}=\sqrt{14}$.

(i)

First, we can note that we can let $\mu=0$ and $\lambda=1$ for convenience since the vector equation holds for all values of $\lambda , \mu$. Then we have $\textbf{r}=\left( \begin{array}{c}6\\t\\2\end{array}\right)$. Putting this into the given equation in (i), we have $12+3t+2=5$, hence $t=-3$.

(ii) Let M be the midpoint of AB. Note that M will lie on the plane $\pi_1$! By the Midpoint Theorem,

$\vec{OM}=\frac{1}{2}[\vec{OA}+\vec{OB}]=\left( \begin{array}{c}4\\-6\\7+\frac{1}{2}s\end{array}\right)$.

Since M lies on the plane, $\vec{OM}\cdot \left(\begin{array}{c}2\\3\\1\end{array}\right)=5$. Expanding the dot product, we get $s=16$.

(iii) We need to show that the normals of the two planes are perpendicular.

$\mathbf{n_1}=\left( \begin{array}{c}2\\3\\1\end{array}\right)$ First we note that B also lies on $\pi_2$. We can obtain a normal of $\pi_2$ by taking the cross product $\mathbf{n_2}=\vec{AB}\times\left( \begin{array}{c}3\\-2\\0\end{array}\right)=\left( \begin{array}{c}4\\6\\-26\end{array}\right)$

We then calculate that $\mathbf{n_1}\cdot\mathbf{n_2}=0$. (shown)

If we draw a diagram, we can see that the shortest distance from A to the line of intersection is actually $|\vec{AM}|$, where M is the midpoint found earlier. $\vec{AM}=\vec{OM}-\vec{OA}=\left(\begin{array}{c}2\\3\\1\end{array}\right)$. Thus $|\vec{AM}|=\sqrt{2^2+3^2+1^2}=\sqrt{14}$.

## Wednesday, 4 February 2015

### Graph Theory Olympiad Question!

January's Math Olympiad Question was on Graph Theory.

There are 2015 points in the space, no three of them are lying on the same line and no

four of them are lying on the same plane. Any pair of points is connected by a segment.

The k-coloring of these $2015 \choose 2$ segments is a coloring of each segment into one of the k

colors so that each color is used at least once. Find the minimal possible value of k for

which any k-coloring contains a triangle with differently colored edges.

Sounds interesting? Check out the Question and Solution at: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm

How to learn more about Graph Theory? Graph Theory is usually taught in university Math / Applied Math courses. Computer Science courses will most probably learn some graph theory too. It is a highly interesting topic that is "elementary" in a sense that it does not use deep theorems, but some counting and combinatoric principles. However, the simplicity is deceptive and Graph Theory has some highly challenging problems. Do check out this book The Fascinating World of Graph Theory by some highly regarded authors like Gary Chartrand (expert in Graph Theory). The author Arthur Benjamin is known for being a very interesting author who also specializes in Mathematical Magic Tricks.

There are 2015 points in the space, no three of them are lying on the same line and no

four of them are lying on the same plane. Any pair of points is connected by a segment.

The k-coloring of these $2015 \choose 2$ segments is a coloring of each segment into one of the k

colors so that each color is used at least once. Find the minimal possible value of k for

which any k-coloring contains a triangle with differently colored edges.

Sounds interesting? Check out the Question and Solution at: http://www.fen.bilkent.edu.tr/~cvmath/Problem/problem.htm

How to learn more about Graph Theory? Graph Theory is usually taught in university Math / Applied Math courses. Computer Science courses will most probably learn some graph theory too. It is a highly interesting topic that is "elementary" in a sense that it does not use deep theorems, but some counting and combinatoric principles. However, the simplicity is deceptive and Graph Theory has some highly challenging problems. Do check out this book The Fascinating World of Graph Theory by some highly regarded authors like Gary Chartrand (expert in Graph Theory). The author Arthur Benjamin is known for being a very interesting author who also specializes in Mathematical Magic Tricks.

## Monday, 2 February 2015

### The Math of Rock, Paper, Scissors!

Who knew the simple game Rock, Paper, Scissors had mathematical implications, and could be found in nature!

Hannah Fry discusses the Math behind the ecological balance of lizards, which turns out to be a natural example of Rock, Paper, Scissors!

There is also a psychological and logical "strategy" of playing Rock, Paper, Scissors shown in the video below.

Hannah Fry discusses the Math behind the ecological balance of lizards, which turns out to be a natural example of Rock, Paper, Scissors!

There is also a psychological and logical "strategy" of playing Rock, Paper, Scissors shown in the video below.

Finally, who would have thought that there is actually a Rock, Paper, Scissors championship! The world champion is a guy called Bob "The Rock"!

Do check out this book Growing Patterns: Fibonacci Numbers in Nature based on another ubiquitous pattern in nature, the Fibonacci Numbers.

## Wednesday, 28 January 2015

### How to Install Mario Games on TI-84 Plus Graphic Calculator?

The TI-84 is a really powerful calculator, capable of graphing and complex calculations. Wait... There is more! It can even install games like Mario!

First, you will need a cable to connect your TI-84 (Plus Pocket SE is what most JC students use in Singapore) to the computer.

Now, to enter the game, click on Apps and select MirageOS. If everything goes well, you should enter MirageOS. Click Enter to search for Mario and you should enter the Mario Game for TI-84+. Press "2nd", which is equivalent to the "A" button on Gameboy, and you should be ready to start. Have fun!

To quit, press "clear" to reach MirageOS, then press "mode".

Note, the Mario game can be found under "prgm", but accessing it there does not work. It will give either "ERR: ARCHIVED" or "ERR: SYNTAX".

To delete Mario from your calculator, go to 2nd->mem->Mem Mgmt/Del. You may remove Mario from the calculator once you reach there.

Want to be a true master at the TI-84 Calculator? Check out this book Ti-84 Plus Graphing Calculator For Dummies! Get up-to-speed on the functionality of your TI-84 Plus calculator Completely revised to cover the latest updates to the TI-84 Plus calculators, this bestselling guide will help you become the most savvy TI-84 Plus user in the classroom!

First, you will need a cable to connect your TI-84 (Plus Pocket SE is what most JC students use in Singapore) to the computer.

### Steps to Install Mario on TI GC

- Go to http://tiwizard.com/downloads/mirage-os-1-2/ and install MirageOS. Once you download it, right-click send to TI Device. You may send it to Archive.
- Next, go to http://tiwizard.com/downloads/mario-2-0/ and install Mario 2.0, the engine for the Texas Instrument Mario Game. Similarly, once you download it, right-click and send to TI Device, under Archive.
- Finally, go to http://tiwizard.com/downloads/mario-2-0/ and install Super Mario Land Level Pack. There are eight levels in total.

Now, to enter the game, click on Apps and select MirageOS. If everything goes well, you should enter MirageOS. Click Enter to search for Mario and you should enter the Mario Game for TI-84+. Press "2nd", which is equivalent to the "A" button on Gameboy, and you should be ready to start. Have fun!

To quit, press "clear" to reach MirageOS, then press "mode".

Note, the Mario game can be found under "prgm", but accessing it there does not work. It will give either "ERR: ARCHIVED" or "ERR: SYNTAX".

To delete Mario from your calculator, go to 2nd->mem->Mem Mgmt/Del. You may remove Mario from the calculator once you reach there.

Want to be a true master at the TI-84 Calculator? Check out this book Ti-84 Plus Graphing Calculator For Dummies! Get up-to-speed on the functionality of your TI-84 Plus calculator Completely revised to cover the latest updates to the TI-84 Plus calculators, this bestselling guide will help you become the most savvy TI-84 Plus user in the classroom!

## Monday, 26 January 2015

### NUS Math Talk: Mathemusical Conversations: Mathematics and Computation in Music Performance and Composition (13 - 15 February 2015)

For those interested in the union of the two diverse subjects Mathematics and Music, do check out this symposium by NUS:

**Mathemusical Conversations**Organizing Committee |

*Original Protagonists*- Louis Chen (National University of Singapore)
- Bernard Lanskey (National University of Singapore)
- Bernard Tan (National University of Singapore)

*Program Chairs*- GĂ©rard Assayag (Institut de Recherche et Coordination Acoustique/Musique)
- Elaine Chew (Queen Mary University of London)

*Local Organizing Committee*- Jenny Ang (National University of Singapore)
- I-Shyan Tang (National University of Singapore)
- Rachel Tang (National University of Singapore)
- Craig de Wilde (National University of Singapore)

Overview |

Mathemusical Conversations is an international workshop bringing together world experts and emerging scholars in and across mathematics and music, with a special focus on mathematical and computational research in music performance and composition that serve as the foundation for understanding and enabling human creativity and for future music technologies.

Read more at: http://www2.ims.nus.edu.sg/Programs/015wmusic/index.php

This symposium reminds me of the book Science and Music (Dover Books on Music) by Sir James H. Jeans, the quintessential book on the intersection of Science and Music. Read more about this wonderful and amazing book here.

## Saturday, 24 January 2015

### Hilarious Math Video

What is 25 divided by 5?

Watch this humorous video on what can go wrong when dividing!

Do also check out this very interesting book titled "Introductory Calculus For Infants". It is a humorous math book about the storybook adventure of two friends as they explore the wonders of calculus.

## Friday, 23 January 2015

### Singapore Math Books

If you are interested to read more about Singapore Math, check out this site on Singapore Math Books!

### The Mystery of the Sine of Nines: sin 999=sin 9999 ?

Today while playing with my calculator, I discovered a curious fact:

sin(999)=-0.987688

sin(9999)=-0.987688

sin(99999)= -0.987688

Note:

In fact, the sine of any number of nines (more than 3), always led to the same number!

This may not work with other digits, for example "8":

sin(888)= 0.20791

sin(8888)=-0.92718

sin(88888)=-0.52992

As a math tutor, definitely I was curious about the mathematics behind this phenomenon. If you want to try to unravel the mystery, do give it a try before reading the answer!

###

The mystery is pretty straightforward once we notice the following:

$\sin (9999^\circ)=\sin (999^\circ+ 25(360^\circ))$

As we know, adding 360 degrees to an angle doesn't affect the result of its sine, since $\sin (x+360^\circ)=\sin (x)$. sin(9999) is actually sine of 25 times of 360 added to 999, hence they are essentially the same value!

Thus, sin(9999)=sin(999).

We can then proceed to show sin(99999)=sin(9999) in a similar way. This will keep on working since 9000=25x360 is already a multiple of 360, hence 9000...000 (more than 3 zeroes) will also be a multiple of 360!

This concludes the mysterious case of the Sine of Nines (it rhymes!).

Trigonometry is a really fun subject. But could it be taught better? Trigonometry often leads to nasty irrational numbers, for example sin(60) is already an irrational number ($\sqrt{3}/2$). Professor Wildberger, author of Divine Proportions: Rational Trigonometry to Universal Geometry argues that there is a better way to present Trigonometry, via the very novel (most people haven't heard of it, let alone seen it) Rational Trigonometry. I have followed his videos on YouTube, and personally it is an interesting idea. With Rational Trigonometry, irrational numbers (which are highly problematic if one thinks about them deeply) are banished, and we can only work with rational numbers.

Check out the book here:

sin(999)=-0.987688

sin(9999)=-0.987688

sin(99999)= -0.987688

Note:

**All angles in degrees.**In fact, the sine of any number of nines (more than 3), always led to the same number!

This may not work with other digits, for example "8":

sin(888)= 0.20791

sin(8888)=-0.92718

sin(88888)=-0.52992

As a math tutor, definitely I was curious about the mathematics behind this phenomenon. If you want to try to unravel the mystery, do give it a try before reading the answer!

###
__The Mystery of the Sine of Nines__

The mystery is pretty straightforward once we notice the following:$\sin (9999^\circ)=\sin (999^\circ+ 25(360^\circ))$

As we know, adding 360 degrees to an angle doesn't affect the result of its sine, since $\sin (x+360^\circ)=\sin (x)$. sin(9999) is actually sine of 25 times of 360 added to 999, hence they are essentially the same value!

Thus, sin(9999)=sin(999).

We can then proceed to show sin(99999)=sin(9999) in a similar way. This will keep on working since 9000=25x360 is already a multiple of 360, hence 9000...000 (more than 3 zeroes) will also be a multiple of 360!

This concludes the mysterious case of the Sine of Nines (it rhymes!).

Trigonometry is a really fun subject. But could it be taught better? Trigonometry often leads to nasty irrational numbers, for example sin(60) is already an irrational number ($\sqrt{3}/2$). Professor Wildberger, author of Divine Proportions: Rational Trigonometry to Universal Geometry argues that there is a better way to present Trigonometry, via the very novel (most people haven't heard of it, let alone seen it) Rational Trigonometry. I have followed his videos on YouTube, and personally it is an interesting idea. With Rational Trigonometry, irrational numbers (which are highly problematic if one thinks about them deeply) are banished, and we can only work with rational numbers.

Check out the book here:

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