Introduction to Trigonometry!

[AngelWings]

EDIT: Just realised Cosine posted an hour ago (How NOT to Memorise Exact Values! - Trigonometry). Hm... well, actually he's posted pretty much the complement and not a lot of it has overlapped, so my efforts haven't gone to waste. Yay!

Memorising Tips for Exact Values
Choose whichever way works best for you. Please feel free to amend (I didn't check this very well.) or post your ways of memorising them, if you feel it is going to be beneficial.

Let's begin by reminding ourselves of the exact values.
Exact Values
They are often listed by resources in the form a table, which you can see right here.

Method 1: Memorising through Repetition and Practise

Spoiler

The most basic method is to just keep using them and do tonnes of questions requiring your exact values. Textbooks usually have quite a large collection of questions of this sort. Seeing it being used in different, but similar, contexts will often make it easier to recall.

Method 2: Exact Values Triangles

Spoiler

Some people are visual learners. This is one way that'll satisfy those people and usually favoured by teachers (or at least in my experience, this statement is true).
There are two triangles that you must memorise in order to take this route. People usually call them the "exact value triangles" for obvious reasons.
The 30^o and 60^o Triangle

Spoiler

The first triangle is equilateral with sides of 2 units and, being equilateral, will have 180/3 = 60^o angles. If we break the triangles in half, we get two equal right-angled triangles of sides 2 units (the hypotenuse), 1 unit (the bottom side, opposite the 30^o) and the square root of 3 units(the left side, opposite the 60^o).
Okay, let's pause. Here's a question for you: Where did these sides come from?

• The 2 units side.
  Spoiler

  The 2 units comes directly from the equilateral triangle. It is the length of one of the equilateral triangles' sides.
• The 1 unit side.
  Spoiler

  This is the side that was halved in the equilateral triangle, so the length of the original equilateral triangle, which was 2 units, is now halved, i.e. 1 unit.
• The square root of 3 side: If you don't know where the square root of 3 comes from, you should try using Pythagoras' theorem (spoiler).
  
  Spoiler

  Pythagoras's theorem can be applied only for right-angled triangles, which we have. The formula is: a²+b²=c² (or h², if you want to call it "h" for hypotenuse, but in this case, let's just go with c²). The hypotenuse is equivalent to the length of c. The pro-numerals "a" and "b" refer to the other two sides.
  Substitute a=1 (see dot point number 2) and c=2 units (see dot point number 1).
  You should receive: 1²+b²=2². What we now want is to get "b" as the subject. So a little rearrangement gives: b²=4-1 and therefore b=square root of 3 units. That's how we got square root of 3 as the side.

The 45^o Triangle

Spoiler

This is a right-angled, isosceles triangle with sides 1 unit, 1 unit and the square root of 2 units (hypotenuse).
Again, question: Where does the hypotenuse's length come from?

Spoiler

It can be worked from Pythagoras' theorem. Substitute a=1 unit and b=1 unit to find c's length (the hypotenuse).

Alternatively, you can see the exact value triangles in pictorial version:30 and 60 degree triangle and 45 degree triangle.
As I have stated in my previous post (see page 1), we can only apply SOH CAH TOA to right-angled triangles. Since they are, you can easily apply these ratios on these triangles to receive your exact values.
A breakdown of the steps required:

Spoiler

• Note which triangle you must use and whether you want to work out sin, cos or tan.
• Substitute values for theta and the sides in the ratio.

Let's try an example. What is the exact value of tan(60^o)?

Spoiler

1. Use the 60^o  triangle
2. apply tan(theta)=opposite/adjacent and substitute values.
3. receive tan(60^o)= sqrt(3)/1= sqrt(3) units

Method 3a: Knowing Consecutive Numbers and Memorising Tangent Values Separately

Spoiler

This is the one that works for me. It's more difficult to prove, but it's kind of a weird pattern that occurs with sin and cos.
Somehow, when you memorise the table, you will find that all the sin values increase consecutively from 0 to 4 and then the square root is taken and finally divided by 2. That is:

Spoiler

sin(0^o)=sqrt(0)/2=0
sin(30^o)=sqrt(1)/2=1/2
sin(45^o)=sqrt(2)/2=1/sqrt(2) (rationalised)
sin(60^o)=sqrt(3)/2
sin(90^o)=sqrt(4)/2=2/2=1

It is the converse for cos values i.e. from 4 to 0, since they are complementary.
For tan values, you can memorise them separately.

Method 3b: Knowing Consecutive Numbers and Knowing that Tangent Values Increase in Quadrant 1

Spoiler

This is essentially the same method as Method 3a.
If you see the graph of tan(x), you will note that from 0 to 90^o, it increases from 0 to positive infinity. For tan values, it works with you memorising the exact values and then putting them in increasing order i.e. 0, 1/sqrt(3), 1, sqrt(3) and infinity. These will then pair up with the angles in increasing order respectively. In other words: tan(0^o)=0, tan(30^o)=1/sqrt(3), tan(45^o)=1 and so forth.

EDIT 2: Method 4: The Finger or Hand Trick (Alchemy's suggestion)

Spoiler

As suggested by Alchemy, there is also the hand trick. I, personally, have only encountered this once and didn't remember it until it was mentioned. (Thanks Alchemy.) Yeah... so I'm obviously not the best person to explain it. Instead I decided to get you guys some resources from people who can.

Helpful resources to learn/ recall the hand trick:
Trigonometric Finger Trick PDF
Unit Circle Hand Trick
Simple Hand Trick for Memorizing Exact Values (A 5 and a half minute video for any AN visual learners out there.)

Applying This to Other Quadrants
You can apply these exact values in any quadrant, as long as you can remember whether the value is positive or negative for that quadrant and if that angle's equivalent (axis of symmetry). Particularly tricky are the negative angles, because it makes you think in the opposite direction, but don't fret, a little practise will make you better at them.
Also, if you need to solve them in radians, it's the same, except you need to remember the conversion between degrees and radians.
Question: How do we convert...
a. from degrees to radians?

Spoiler

Multiply the amount of degrees by pi/180.

b. from radians to degrees?

Spoiler

Multiply the amount of radians by 180/pi.

Note: I've never done extensive notes like these before on here, so I'm open to constructive criticism. I apologise for my lack of LaTeX and computer knowledge and used the most simple method to cater for the amount of time I had to write this.

[cosine]

Go AngelWings! Good job on the explanations!

Here is my How NOT to Memorise Exact Values! - Trigonometry, the above methods is absolutely easier to remember, but the way I show it may help you remember them visually. Hope you guys enjoy

[AngelWings]

Go AngelWings! Good job on the explanations!

Here is my How NOT to Memorise Exact Values! - Trigonometry, the above methods is absolutely easier to remember, but the way I show it may help you remember them visually. Hope you guys enjoy

Yeah, thanks Cosine. I would have gone visual, but time constraints.
Anyway, Cosine's done a great job explaining it visually.

(200 posts)

[MathsNerd203]

Could you do one of these for calculus, probability and matrices?

[cosine]

Could you do one of these for calculus, probability and matrices?

Of course! I will be doing calculus next, thanks for the suggestion!

[alchemy]

someone show 'em the hand trick as well for remembering exact values
might be useful up to the point when they become etched into your head anyway...

[AngelWings]

someone show 'em the hand trick as well for remembering exact values
might be useful up to the point when they become etched into your head anyway...

Just edited post. Consider that done.

[achre]

What an apt username. Great work!

[cosine]

What an apt username. Great work!

Haha yeah! Thanks man

[Arithmetic]

Great work cosine!!!

[cosine]

Great work cosine!!!

No worries buddy, glad it helped!

[cosine]

Here is my third part to the Trigonometry series

How to Sketch Circular Functions Easily!