1. Trigonometry Lesson 1: Introduction
Hello again! Today we'll be finding out two things: what trigonometry is, and how
it relates to triangles. Now, before you sigh and exit the page, listen to what I have to
say. Trigonometry isn't as boring as it sounds – although some of the later topics are
more exciting.
From Wikipedia: Trigonometry (from Greek trigōnon “triangle” + metron
“measure”) is a branch of mathematics that studies triangles and the relationships
between their sides and the angles between these sides. This is a pretty good definition.
We'll be dealing with triangles quite a bit, but what may surprise you is how these
relationships between a triangle's sides and angles are applicable to real-life situations. I
can think of countless examples in physics, for example, that heavily rely trigonometric
functions (I'll just call them trig functions for short). Just imagine, without these
functions, we wouldn't even be able to solve basic physics problems. Then where would
we be?
My friend (who has never taken the class) once told me that trig “is basically
geometry on steroids.” I would disagree with this. While they are somewhat similar and
some topics overlap, trig is not very much like geometry at all. Much less emphasis is
placed on shapes, and instead on angles and graphs. Don't worry though; they aren't hard
if you know what you're doing.
Alright. That's enough talking about trig, now how about we get to learning? Here
we go:
Properties of Triangles
Before I get into what the functions are, I should briefly mention
some things about triangles. A triangle is a shape with three straight
sides and angles that add up to 180°. It also must be a closed shape
(hence, it is a polygon). For example, the figure to the right is NOT a
triangle. One of its legs doesn't reach far enough to close the gap.
That brings me to another topic: triangle terminology. A right
triangle is any shape that meets the criteria above and has ONE angle
that is 90° (called a right angle). Right angles are denoted by a small square like in the
figure below. A triangle can have at most one right angle because two would make it 90°
+ 90° = 180°; there's no room left for the third angle. I will list some other terms you
should be familiar with. If any of this is foreign to you, you should watch some of the
videos in Khan Academy's geometry playlist.

2. • Equilateral triangle – triangle with all equal sides and
angles
• Isosceles triangle – triangle with two equal sides and two
equal angles
• Scalene triangle – triangle with no equal sides or angles
• Obtuse triangle – triangle with one angle greater than 90°
• Acute triangle – triangle with all angles less than 90°
This should be common knowledge to you. It is also worth noting that a triangle
can have more than one of the properties listed; a triangle can be both scalene and
obtuse, or an isosceles triangle can be acute.
I would like to point something out about the rather colorful triangle I made.
Obviously, it has one right angle and meets the criteria for being a triangle (we'll assume
the angles add up to 180°). Therefore, it is also a right triangle. Don't be fooled – just
because the figure is slanted does not mean it's not a triangle. The blue and green sides
are called the legs of the right triangle. The red side is called the hypotenuse. The
hypotenuse is the side opposite the right angle (again, denoted by the square). It is
always the longest side of a right triangle. The legs are the two sides that aren't the
hypotenuse. Make sure you understand this terminology, because it's vital to your
understanding of the rest of the course. Also know that legs and hypotenuses only apply
to right triangles. If you're talking about any other type of triangle, it doesn't make sense
to mention those things.
Okay, now I think I can actually get into the trig functions. Hopefully that was a
breeze for you. And it should have been – that's just basic geometry.
Basic Trig Functions: Sine, Cosine, Tangent
Note: If at any point you don't understand the material, please skip to the end of
the PDF where I will link to a Khan Academy video that explains the topic
beautifully. Then come back to this.
You may have heard of these functions before. They are sine (usually written as
sin, pronounced like the word 'sign'), cosine (written as cos, pronounced like 'cosign'),
and tangent (written as tan, pronounced like 'tan' [as in the darkening of skin]). These
are crucially important, so make sure you remember them. Now, what are they?
I'll start with the definition of sin. Be patient – this won't make sense until I show
you a picture. Sin(A) is the side opposite of the angle A divided by the hypotenuse
(obviously, we are working with right triangles). Let me show you what I mean.
We'll call the legs of a triangle x and y. The hypotenuse is z (I'm just arbitrarily
assigning variables). Take a look at the picture below.

3. Note that, from the little white square, we can tell this is
a right triangle. The symbol θ is used to denote the angle
in the top left corner. This symbol is called theta. It's a
Greek letter commonly used for angles, much like how x
is used as a variable for the unknown in algebra. Don't let
this confuse you – it's just a variable that we're using to
represent an angle. Got that? Okay, now let's see how sin
fits into all of this.
I told you that sin(A) = the side opposite to A /
hypotenuse. The forward slash means division. It's
important to realize that the A is an ANGLE. You can't use the sin function on anything
but an angle. Let's apply that rule to theta (θ).
sin(θ) = side opposite to θ / hypotenuse
sin(θ) = y/z
That was easy, wasn't it? You just look at whichever side
is opposite to the angle in question, and divide that by
the hypotenuse. That gives you the sin of an angle in a
right triangle.
What if I changed θ to be the angle in the bottom right?
What would be sin(θ) then? To avoid confusion, I'll call
the bottom right angle angle B.
sin(B) = side opposite to B / hypotenuse
sin(B) = x/z
As we can see, the sin function changes depending on
what angle we call it on. For the angle in the top left,
sin(θ) was y/z. But for sin(B), it's x/z. You may be
asking yourself, what's sin(90°)? Well, you'll find out
soon enough, but for this lesson, you should know that you will NEVER be asked to call
trigonometric functions on the right angle. This does not mean that you won't be asked
to find what trig functions of 90° are.
Ignore that last bit if it confused you. So, in summary, sin of an arbitrary angle
theta = opposite/hypotenuse. Great, but what about cos and tan?

4. Cos follows a similar definition. Cos(θ) = adjacent side/hypotenuse. Adjacent
means next to. We call the adjacent side the side that is next to an angle and that
isn't the hypotenuse. The figure below shows how to find the cos of an angle we'll call
α. This is another Greek letter called alpha, also frequently used to represent angles.
Instead of x, y, and z for sides, let's use d, e, and f. I like a little variety.
cos(α) = adjacent/hypotenuse
cos(α) = d/f
For practice, what would sin(α) be?
Well, take the side opposite and divide it by
the hypotenuse. You should get e/f.
Look back at the pictures in the last page.
What is cos(θ) and cos(B)? The answers will be at the end of this PDF. Try to find the
cos and sin of all angles (excluding the right angle) in the following triangle. The
answers will also be at the end.
1. sin(x) = ?
2. sin(y) = ?
3. cos(x) = ?
4. cos(y) = ?
Tan is also very similar. The only difference is that it
doesn't involve the hypotenuse. Tan(θ) =
opposite/adjacent. You follow essentially the same
process. Just remember that the adjacent side CANNOT
be the hypotenuse.
tan(A) = opposite/adjacent
tan(A) = o/k
tan(B) = opposite/adjacent
tan(B) = k/o
Remember that the functions change for each angle. tan(A) (which the figure was drawn

5. for) is o/k, but tan(B) is k/o. I keep switching between variable names just to show you
that it doesn't matter what you call the sides and angles. I encourage you to find the tan
of angles in previous figures; practice is key.
To illustrate that these trig functions only work for right
triangles, take a look at this picture and try to find sin(θ). You can
find the opposite side, sure, but what's the hypotenuse? Problem:
hypotenuses only exist for right triangles. The same problem
occurs if you try to find cos(θ), with the additional issue of not
knowing what the adjacent side is. There are two adjacent sides. In
right triangles, we call the adjacent side the side that is next to an
angle and that isn't the hypotenuse. Again, no hypotenuse here.
Think about tan(θ). Can you find it? (remember: tan is opposite/adjacent)
There is, however, a way to use sin to solve most triangles (even those that are not
right). We will be getting to that later in the course.
Tying this all Together
That's a lot to memorize, huh? It is, and very few people memorize trig functions
like that. There's a helpful mnemonic for this:
SOH CAH TOA
This is definitely the easiest way to remember how each function is defined.
SOHCAHTOA. Memorize it. Live it. Breathe it. You need it.
S = O/H
C = A/H
T = O/A
As you can see, sin = o/h, cos = a/h, and tan = o/a. O stands for opposite, h stands
for hypotenuse, and a stands for adjacent, but you probably figured that out already.
You'll never forget the trig functions with this.
Conclusion
If you understood and mastered this lesson, bravo! You are on your way to
becoming a trig master. The next few lessons should be pretty easy if you understand
this material well. I encourage you to do all of the practice problems in this lesson (if
you didn't already) and the ones I link to. Of course, I can't make you do them, but you
should know that what you put into this course is directly proportional to what you get

6. out (see what I did there?).
I also recommend watching the Khan Academy video, even if you understood
everything I had to say. More practice and exposure to the topic can only help you.
Hope you enjoyed the lesson.
~ Dan