Surviving Trigonometry

Hewwo!

I've been AFK on advice-column lately. Or longly. Let's change that.

A couple weeks before exam season, it's quite alarming that many people have ZERO clue what the fuck they're doing. Especially in trigonometry territory.

Wait, who am I and why should you listen to me?

For those who are new to the IB server and/or the website, I hate trigonometry. I really do. I hate it so much that every exam season I have to rope my friend CreativeUsername into doing a revision session with me, on trigonometry, on the IB server. He never ceases to make fun of my toxic relationship with trigonometry. I have proofs.

Trigonometry is easy... if you know what you're doing. Luckily this is where most people are dragged down. Warren Buffett once said "Be greedy when the market is fearful and be fearful when the market is greedy", you'll reap the most when acing the topic that most people want to deny its existence.

As a proud survivor, I'm gonna lay all my cards on the table. No secrets here. Follow my path and I can guarantee you won't be clueless in trigonometry lessons.

The foundations

Trigonometry is very different from everything you've seen in maths. It's a different breed.

Many students fear that their gaps in geometry will hold them back in trigonometry.

I agree with that claim... 6.9%.

93.1% of the time you don't need to use fancy geometry tools to know what's going on. This is a chance for you to start fresh.

The one thing thou shalt not forget: angles on a straight line add up to 180 degrees.

Thank you for coming to my ted talk.

Roadmap to success

This is the barebone of your trigonometry journey in IB.

Once you have these big boys down, you can learn the small things without much fuss.

Pythagorean theorem

Famously known as the 3-4-5 or 5-12-13 combo.

THEY ARE NOT PHONE NUMBERS, I repeat!

Once you have two sides of a right triangle, it's easy to find the missing one.

Right angle trigonometry

Focus on knowing when to use each of the three ratios - sine, cosine and tangent.

Many teachers jump right into the deep end teaching the three ratios. I remembered being very fucking confused in grade 9 when our teacher just said "get on with it!". Painful experience. So now I aspire to make it less painful for you.

In my trigonometry booklet (WIP) I spend quite a bit of time on practise identifying the hypotenuse, the opposite and the adjacent.

• the hypotenuse is always opposite the right angle
• the opposite is, surprise surprise, always opposite the angle you're talking about
• the remaining side is the adjacent

When you're done with this boring stuff (take your time!) let's take it a notch further.

What if we have a non-right triangle? Panic? Cry? 

No. You're stronger than that. Grow up.

Non-right triangles

Now we extend what we've seen in right triangle trigonometry.

• The sine rule - an extension of the ratios
• The cosine rule - an extension of the Pythagorean theorem
• Area of a triangle - the odd child in the family

Collectively I call them "the trio". 

Whenever you go to a Chinese restaurant you're always asked to pick a main dish, a side dish and a drink from a bazillion options. This is similar - don't be surprised if you have to use a handful of them in the same question!

After this is all done and sorted, ladies and gentlemen, allow the fun to begin.

The unit circle

Everything we've done so far are limited to a triangle. All angles between 0 and 180. What if we want to be adventurous and break out of the limits, in either direction? The unit circle gives you everything you ever wanted.

Massive advice: pay very close attention to symmetry. You'll notice some interesting patterns. And soon enough you'll see them EVERYWHERE.

And it's right here.

Trigonometric functions

If you've made it this far, kudos.

This is where shit starts going south for many people. Why, you may ask. Because they didn't take my words seriously and don't really understand the unit circle.

George Santayana wrote: "Those who cannot remember the past are condemned to repeat it." More commonly known as "history repeats itself".

I've put all my cards on the table. Now it's your job to see if what I'm saying is life changing or total bollocks.

5 seconds ago I mentioned "symmetry". This is where it kicks in.

Exploiting symmetry is the QUICKEST way to make your life easier and avoid PAGES of computation.

Useful checklist

What information must you extract from a trig function?

• Maximum and minimum values
• Period
• Horizontal shifts

The three things you must keep an eye on. The rest can be readily calculated.

Once you could imagine the graphs of trig functions in your head and put them on paper, solving trig equations and inequalities is a piece of cake.

I've discussed this in my trigonometry booklets which you can find here and here.

Lastly, and the most formidable.

Trigonometric identities

Headaches. Brain damage. Head scratching.

Easily the most beloved topicin trigonometry that has a proven record of students bombing left right and centre.

If you're taking AAHL and your teacher is a mathematician at heart, you'll prove these identities in many ways. A purely geometric proof, tick. A complex number proof, tick.

But if you're like most people who couldn't care less about proofs, you can always reason your way through using the formula booklet. 

These words probably don't make sense to you now but it will.

When in doubt, convert everything to sine and cosine.

Yes, it's tedious. Yes, it's long. Yes, it works.

If you're struggling with this, I've also written a booklet just on identities.

If you took my advice seriously, you should be very comfortable converting sine to cosine and vice versa. You have to, if you finish all the questions in my booklets. Otherwise they'll be coming for you for committing a sin(e).

This joke obviously works better than paper lmfao but we'll keep it there.

Of course, this means your trig equations are now elevated to a new level. Be grateful you're doing IB and not the local Asian curriculum - we go real deep into the bazillion forms of trig equations that has absolutely zero practical use in real life.

So there you go. That's how I navigated through trigonometry.

You don't have to be a master in geometry to do well in trig.

You don't have to be good at maths to do well in trig - I've debunked this at the beginning.

Everyone can do trig if they have enough trigs under the belt. :D

List of links

Collection of unit tests and booklets 

Last words

If you need help with trigonometry, or just maths in general, you can find me on the IB Discord. 

If you need more serious help, I'm available for tutoring. Just DM me for more details and we'll discuss what's the best option for you going forward!

And lastly, if you find this post or the booklets useful, please spread the word or consider buying me a coffee here.

Keep an eye on this blog over the summer as I'll be churning out more advice column like this!

Have fun y'all and may the odds be ever in your favour.

Andrew