Main content

## Geometry (all content)

### Course: Geometry (all content) > Unit 13

Lesson 2: Introduction to the trigonometric ratios# Intro to the trigonometric ratios

Sin, cos, and tan are trigonometric ratios that relate the angles and sides of right triangles. Sin is the ratio of the opposite side to the hypotenuse, cos is the ratio of the adjacent side to the hypotenuse, and tan is the ratio of the opposite side to the adjacent side. They are often written as sin(x), cos(x), and tan(x), where x is an angle in radians or degrees. Created by Sal Khan.

## Want to join the conversation?

- What do you use for the adjacent side if you're trying to figure out the sin cos and tan for the right angle?(209 votes)
- Well, in beginning trigonometry, it's convenient to evaluate sin/cos/tan by using soh-cah-toa, but later, as you get into the unit circle and you start taking taking stuff like sin(135) and tan(-45) you don't use the adjacent-opposite-hypotenuse much anymore. If you can think of it intuitively, though, sin(90) means that the opposite side is infinitely long, and the hypotenuse is also infinitely long, so sin(90)=1. cos(90) means adjacent over the hypotenuse, which is infinitely long given that the angle is 90 degrees, so any number over infinity is 0, so cos(90)=0. tan(90)=sin(90)/cos(90)=1/0, so tan(90) doesn't exist.(113 votes)

- How can you figure out which is the opposite or the adjacent?(104 votes)
- The opposite side is the side opposite of the angle that you are trying to solve for. The adjacent side is the side next to the angle you are solving for.(74 votes)

- A small question. We name the ratios as "Sine", "Cosine", "Tangent", "Cotagent", "Secant" and "Cosecant". The last four can be drawn of circle. Is "Sine" also a part of circle. I mean can it be drawn on circle like tangent and secant. I know its a useless question, but I was just wondering. Thanks for your time.(2 votes)
- I think that's a great question! This is a pretty cool story (to me at least). The word that the Arabs used for sine was the same as their word for "chord", but when a European translated it into Latin he read it wrong and translated it as sinus, which is the Latin word for chest. So it's a historical accident that secant and tangent have geometric meanings but sine doesn't.(117 votes)

- Why does Sal (the person talking in the video) use theta or some other greek letter for the angles instead of a normal variable, like x or y, for
*every angle*he shows the sin, cos, and tan for?(5 votes)- The reason is because in the world of math (not khan academy's "world of math"), mathematicians usually use x and y for missing lengths, and use Greek letters for unknown angles, most likely in honor of Elucid, founder of geometry, who was Greek.

Hope that helps!(27 votes)

- My question is around
**how**to calculate sin, cos or tan. I've pushed the sin/cos/tan button many times on my calculator with*no*_idea_ what is actually happening. From doing some of my own research, it seems like a Taylor Series may have to be used? Is this the only way? Did someone once sit down and measure every angle and every side of the triangle to get each ratio into a large table? Would it then be something like a look up table with the calculator simply searching for the closest ratio that matches what is typed into the calculator?(12 votes)- sin cos and tan are basically just functions that relate an angle with a ratio of two sides in a right triangle. Sin is equal to the side opposite the angle that you are conducting the functions on over the hypotenuse which is the longest side in the triangle. Cos is adjacent over hypotenuse. And tan is opposite over adjacent, which means tan is sin/cos. this can be proved with some basic algebra.(5 votes)

- To help you guys understand SOH CAH TOA even better, I decided to create this comment as a question.

SOH: [S is Sine, O is Opposite, H is Hypotenuse]. Then, [Sine= Opposite/Hypotenuse].

CAH: [C is Cosine, A is Adjacent, H is Hypotenuse]. Then, [Cosine= Adjacent/Hypotenuse].

TOA: [T is Tangent, O is Opposite, A is Adjacent]. Then, [Tangent= Opposite/Adjacent].

To simplify it to make you guys understand even better, knowing the short form for it, I shall show it down below.

SOH: S= O/H

CAH: C= A/H

TOA: T= O/A

Now, since I already told you guys about the SOH CAH TOA form, I shall give you guys an example.

For Example:

If you are given a triangle where the two significantly shorter sides are given and you wish to obtain the longest side termed the hypotenuse, you recognize that it satisfies the terms necessary to use either the SOH or CAH form; You, therefore, proceed to identify which is adjacent and which is the opposite. The opposite, which is clearly identifiable due to its name, is the side that is directly OPPOSITE the given angle. The adjacent is therefore the side which forms a 90° angle to the opposite.

Another Example:

For purposes, the given angle is 45°. So you then proceed to imply due to the SOH form that Sine45= opposite divided by the hypotenuse. For you to obtain the hypotenuse, we transpose it for the hypotenuse to become the subject of the formula. By cross multiplying we obtain the formula:

Hypotenuse = Opposite divided by Sine45.

Note: Some of these contents are copied from @machyl69.

Hope This Helps,

Thank You!(11 votes) - At1:19

what do you call the hypotenuse if it's not a right angle triangle(3 votes)- Only right triangles have a hypotenuse. In a scalene (non-right) triangle, they are all just called sides.(12 votes)

- How do you use trigonometry on 3d and even 4d shapes and objects?(5 votes)
- Trigonometry can be applied to 3d objects. Consider you have a cube, and you know that angle from cube diagonal to diagonal of square is 45° from here you can easily apply these methods. For 4d would be similiar but yet 4d system would need to be defined and you need to derive many of the equations for that abstraction.(5 votes)

- Is hypotenuse the longest side or what?(0 votes)
- Yes, the hypotenuse is the longest side of a right triangle. It is opposite to the right angle.(7 votes)

- What are the other trig functions? I only remember the cosicant (not spelled right I know).(4 votes)
- cosecant ( csc(Θ) ) in the inverse of sine ( 1/sin(Θ) );

secant ( sec(Θ) ) is the inverse of cosine;

cotangent ( cot(Θ) ) is the inverse of tangent;

There are also six more, but they are a bit hard to describe. They are "arc-" sin, cos, tan, csc, sec, and cot. You might want to check out this video: https://www.khanacademy.org/math/trigonometry/trig-equations-and-identities/inverse-trig-functions/v/inverse-trig-functions-arcsin

He reviews some other things but then talks about arcsine starting at2:08.(5 votes)

## Video transcript

In this video, I want to give
you the basics of trigonometry. And it sounds like a
very complicated topic, but you're going to
see that it's really just the study of the ratios
of sides of triangles. The "trig" part of trigonometry
literally means triangle. And the "metry" part
literally means measure. So let me just give
you some examples here. And I think it'll make
everything pretty clear. So let me draw some
right triangles. Let me just draw
one right triangle. So this is a right triangle. And when I say it's
a right triangle, it's because one of the
angles here is 90 degrees. This right here
is a right angle. It is equal to 90 degrees. And we'll talk about
other ways to show the magnitude of angles
in future videos. So we have a 90-degree angle. It's a right triangle. And let me put some
lengths to the sides here. So this side over
here is maybe 3. This height right
over there is 3. Maybe the base of the
triangle right over here is 4. And then the hypotenuse of
the triangle over here is 5. You only have a hypotenuse
when you have a right triangle. It is the side opposite
the right angle. And it is the longest
side of a right triangle. So that right there
is the hypotenuse. You probably learned that
already from geometry. And you can verify that this
right triangle, the sides work out. We know from the
Pythagorean theorem that 3 squared
plus 4 squared has got to be equal to the
length of the longest side, the length of the hypotenuse
squared, is equal to 5 squared. So you can verify
that this works out. This satisfies the
Pythagorean theorem. Now, with that out of the
way, let's learn a little bit of trigonometry. So the core functions
of trigonometry-- we're going to learn
a little bit more about what these functions mean. There is the sine function. There is the cosine function. And there is the
tangent function. And you write S-I-N,
C-O-S, and tan for short. And these really just specify--
for any angle in this triangle, it'll specify the
ratios of certain sides. So let me just
write something out. And this is a little
bit of a mnemonic here, so something just
to help you remember the definitions of
these functions. But I'm going to
write down something. It's called soh cah toa. And you'll be amazed how far
this mnemonic will take you in trigonometry. So we have soh cah toa. And what this tells
us-- soh tells us that sine is equal to
opposite over hypotenuse. It's telling us-- and this won't
make a lot of sense just yet. I'll do it a little bit
more detail in a second. And then cosine is equal to
adjacent over hypotenuse. And then you finally
have tangent. Tangent is equal to
opposite over adjacent. So you're probably
saying, hey, Sal. What is all this opposite,
hypotenuse, adjacent? What are we talking about? Well, let's take an angle here. Let's say that this
angle right over here is theta, between
the side of length 4 and the side of length 5. This angle right here is theta. So let's figure out
what the sine of theta, the cosine of theta, and what
the tangent of theta are. So if we want to first
focus on the sine of theta, we just have to
remember soh cah toa. Sine is opposite
over hypotenuse. So sine of theta is
equal to the opposite. So what's the opposite
side to the angle? So this is our angle right here. The opposite side, so
not one of the sides that are kind of
adjacent to the angle. The opposite side is the 3. It's opening onto that 3. So the opposite side is 3. And then what's the hypotenuse? Well, we already know. The hypotenuse here is 5. So it's 3 over 5. The sine of theta is 3/5. So if someone says, hey,
what's the sine of that? It's 3/5. And I'm going to
show you in a second that if this angle
is a certain angle, it's always going to be 3/5. The ratio of the opposite
to the hypotenuse is always going to be the same,
even if the actual triangle were a larger triangle
or a smaller one. So I'll show you
that in a second. But let's go through all
of the trig functions. Let's think about what
the cosine of theta is. Cosine is adjacent
over hypotenuse. So remember. Let me label them. We already figured out that
the 3 was the opposite side. This is the opposite side. And only when we're
talking about this angle. When you talk about this angle,
this side is opposite to it. When you talk about this angle,
this 4 side is adjacent to it. It's one of the sides
that kind of make up, that kind of form
the vertex here. So this right here
is an adjacent side. And I want to be very clear. This only applies to this angle. If we were talking
about that angle, then this green side
would be opposite and this yellow side
would be adjacent. But we're just focusing on
this angle right over here. So cosine of this angle--
we care about adjacent. Well, the adjacent side
to this angle is 4. So it is adjacent
over the hypotenuse. It's the adjacent, which is
4, over the hypotenuse-- 4/5. Now let's do the tangent. The tangent of theta,
opposite over adjacent. The opposite side is 3. What is the adjacent side? We already figured that out. The adjacent side is 4. So knowing the sides
of this right triangle, we were able to figure
out the major trig ratios. And we'll see there
are other trig ratios, but they can all be derived
from these three basic trig functions. Now, let's think about another
angle in this triangle. And I'll redraw it just
because my triangle is getting a little bit messy. So let's redraw the
exact same triangle. And once again, the
lengths of this triangle are we have length 4 there,
we have length 3 there, and we have length 5 there. The last example,
we used this theta. But let's do another
angle up here. And let's call this
angle-- I don't know. I'll think of something,
a random Greek letter. So let's say it's psi. I know it's a
little bit bizarre. Theta is what you normally use. But since I already used
theta, let's use psi. Actually, instead of psi,
let me just simplify it. Let me call this angle x. So let's figure out the trig
functions for that angle x. So we have sine of x is
going to be equal to what. Well, sine is opposite
over hypotenuse. So what side is opposite to x? Well, it opens onto this 4. So in this context, this
is now the opposite. Remember, 4 was adjacent to this
theta, but it's opposite to x. So it's going to be 4 over--
now, what's the hypotenuse? Well, the hypotenuse
is going to be the same regardless of
which angle you pick. So the hypotenuse is
now going to be 5. So it's 4/5. Now let's do another one. What is the cosine of x? So cosine is adjacent
over hypotenuse. What side is adjacent to x? That's not the hypotenuse. You have the hypotenuse here. Well, the 3 side-- it's
one of the sides that forms the vertex that x is at,
and it's not the hypotenuse. So this is the adjacent side. That is adjacent. So it's equal to 3
over the hypotenuse. The hypotenuse is 5. And then finally, the tangent. We want to figure
out the tangent of x. Tangent is opposite
over adjacent. Soh cah toa-- tangent is
opposite over adjacent. The opposite side is 4. I want to do it in
that blue color. The opposite side is 4,
and the adjacent side is 3. And we're done. In the next video, I'll do a
ton of more examples of this just so that we really
get a feel for it. But I'll leave you
thinking of what happens when these angles
start to approach 90 degrees, or how could they even get
larger than 90 degrees. And what we're going to see is
that this definition, the soh cah toa definition, takes us
a long way for angles that are between 0 and 90 degrees, or
that are less than 90 degrees. But they kind of start to mess
up really at the boundaries. And we're going to introduce
a new definition, that's kind of derived from the
soh cah toa definition, for finding the sine,
cosine, and tangent of really any angle.