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### Course: High school geometry > Unit 5

Lesson 5: Introduction to the trigonometric ratios# Triangle similarity & the trigonometric ratios

Sal explains how the trigonometric ratios are derived from triangle similarity considerations. Created by Sal Khan.

## Want to join the conversation?

- What do you learn about a triangle from finding the sine, cosine and tangent?(80 votes)
- I wasnt sure that the other answers were really answering yours, they seemed to be more deep. If your question wasn't meant to be deep then I can answer it. When you take the sine, cosine, or tangent of a number you usually get a decimal number. Tangent is different, its usually a bigger number than the others. Now, this decimal number seems useless, i mean what do you do with it? Well, you can use this number to find a missing side length of a right triangle. Say you have all the angle measures but only one side length of a right triangle. You have the length of side A and you need to know the length of side B. Find which one you need sin, cos, or tan and enter it in the calculator. You get the answer so know you multiply the answer times the length of side A and the answer you get is side B!(37 votes)

- Can a cosine be negative? If so, when is it negative and when (if it can be positive) is it positive? Whenever I try to find cosines on my calculator, it is negative. Is that right?(41 votes)
- A cosine can be negative if the angle is more than 90 degrees and less than 270 degrees.

If you are using a calculator, you have to make sure it is set to degrees and not radians. If it is set to radians, you will get the wrong value all the time and you will sometimes get negatives when your answer should be positive.

A simple check to see if your calculator is right is to take cos of 60 degrees. The answer should be 0.5 , if the calculator thought it was 60 radians the answer will be -0.95241298

Hope this helps.(62 votes)

- Do the trigonometric definitions sine, cosine, and tangent apply to any angles of the right triangle? Meaning, can data be the 90 degree angle or can it only be one of the base angles?(5 votes)
- I think by "data", you meant "theta." Yes, the trigonometric ratios can be applied to all angles of a right triangle. When you have a non-right triangle, you will need to use some formulas to apply them.(16 votes)

- How do you determine the degree of an angle? Other than using a protractor! :D(9 votes)
- You can use a trig ratio and 2 of the known measurements, use the law of sines, law of cosines, etc. There are a bunch of ways to find the measure of an angle.(9 votes)

- Why is this in lesson 2? I really think this should be in lesson 1, since it teaches information that is used in lesson 1(13 votes)
- I feel like this is a dumb question, but what is theta?(6 votes)
- theta is a Greek letter which are commonly used for unknown angles, theta is one of the most common letter used(15 votes)

- Sal mentioned in the video that mathematicians gave trignometric ratios names: sine, cosine, and tangent. But of all the names they could've picked in the world, why those three? I don't see any connection between the ratios and the names.(5 votes)
- Oddly enough, trigonometry is really about circles. And, as a result, the names for trig functions come from circles too.

Sine comes from a Sanskrit word meaning "chord". A chord is a segment joining two points on the circumference of a circle.

For cosine, the "co-" stands for complementary. Complementary angles are those that add to 90 degrees. If you take the cosine of an angle and the sine of its complement, you get the same answer. For example, cos(30)=sin(60).

When talking about circles, a tangent is a line that hits one point on a circle. We still use the word "tangent" (besides the trig function tan) today, especially in calculus.

There are three other trig functions that we don't use as often: cotangent, secant, and cosecant. A secant line is a line that intersects two points on a circle. Also, notice the pattern with the "co-". The complementary rule applies to tan/cot and sec/csc too.(16 votes)

- what is difference between similar and congruent triangles?(6 votes)
- If two triangles are congruent, they're exactly the same. However, if they are similar, they look like same but have different size.(7 votes)

- could someone explain what sal is talking about please I don’t quite understand . is there another video i should watch to understand this because i’m new to trigonometry so i haven’t got a clue what i’m doing 😆(6 votes)
- In this video, Sal is explaining how to determine the relationship between two right triangles when they share an angle of the same measure (theta). They explain that if two triangles have two angles in common, the third angle is also the same. Since the sum of the angles of any triangle is 180 degrees, this means that the two triangles are similar. He then goes on to explain that the ratio of corresponding sides of similar triangles is always the same. Using this fact, he derive several equations relating the sides of the two triangles. These equations are true for any right triangle with an angle theta and are the trigonometric functions.(3 votes)

- Why are there not functions to calculate the ratios of angles other than 90°?

e.g.

sinₓ°(θ°) = opposite/hypotenuse of θ° in a x° triangle.

cosₓ°(θ°) = adjacent/hypotenuse of θ° in a x° triangle.

tanₓ°(θ°) = opposite/adjacent of θ° in a x° triangle.

Here we could define hypotenuse as the angle opposite to x°, opposite as the side opposite to θ° and adjacent as the side adjacent to θ° that is not the hypotenuse.

And this should work because of triangle similarity(Euclid's Elements, Book VI, Proposition 4):

angle 1 = x°

angle 2 = θ°

angle 3 = 180-x°-θ°

Establishing a relationship like this would help us solve for angles and sides in non-90° triangles. e.g.:

x° = 60°

θ° = 70°

side adjacent to 70° = x

side opposite to 70° = 5

tan₆₀°(70°) = 5/x

x = 5/tan₆₀°(70°)

Thank you(4 votes)- Law of Sines takes care of that.

sin(𝐴)∕𝑎 = sin(𝐵)∕𝑏 ⇒ 𝑎∕𝑏 = sin(𝐴)∕sin(𝐵)

In other words, the ratio between any two sides in any triangle is equal to the ratio between the sines of their opposite angles.

Given two angles, we easily calculate the third, and thereby we can find any trig ratio we want just using the sine function.

In your example, the angle opposite to side 𝑥 is 180° − (60° + 70°) = 50°, and so

5∕𝑥 = sin(70°)∕sin(50°) ⇒ 𝑥 = 5 sin(50°)∕sin(70°)(4 votes)

## Video transcript

We've got two right
triangles here. And let's say we also
know that they both have an angle whose measure
is equal to theta. So angle A is congruent
to angle D. What do we now know about
these two triangles? Well for any triangle, if
you know two of the angles, you're going to know
the third angle, because the sum of the
angles of a triangle add up to 180 degrees. So if you have two
angles in common, that means you're going to
have three angles in common. And if you have three
angles in common, you are dealing with
similar triangles. Let me make that a
little bit clearer. So if this angle is
theta, this is 90. They all have to add
up to 180 degrees. That means that this
angle plus this angle up here have to add up to 90. We've already used up
90 right over here, so angle A and angle B
need to be complements. So this angle right over here
needs to be 90 minus theta. Well we could use the
same logic over here. We already use of 90
degrees over here. So we have a
remaining 90 degrees between theta and that angle. So this angle is going to
be 90 degrees minus theta. You have three corresponding
angles being congruent. You are dealing with
similar triangles. Now why is that interesting? Well we know from
geometry that the ratio of corresponding sides
of similar triangles are always going to be the same. So let's explore the
corresponding sides here. Well, the side that
jumps out-- when you're dealing with the
right triangles-- the most is always the hypotenuse. So this right over
here is the hypotenuse. This hypotenuse is
going to correspond to this hypotenuse
right over here. And then we could
write that down. This is the hypotenuse
of this triangle. This is the hypotenuse
of that triangle. Now this side right
over here, side BC, what side does
that correspond to? Well if you look
at this triangle, you can view it as the side that
is opposite this angle theta. So it's opposite. If you go across the
triangle, you get there. So let's go opposite angle D. If you go opposite
angle A, you get to BC. Opposite angle D, you get to EF. So it corresponds to this
side right over here. And then finally, side AC
is the one remaining one. We could view it
as, well, there's two sides that make up
this angle A. One of them is the hypotenuse. We could call this, maybe,
the adjacent side to it. And so D corresponds
to A, and so this would be the side
that corresponds. Now the whole reason I
did that is to leverage that, corresponding
sides, the ratio between corresponding
sides of similar triangles, is always going to be the same. So for example, the ratio
between BC and the hypotenuse, BA-- so let me write
that down-- BC/BA is going to be equal to EF/ED,
the length of segment EF over the length of segment ED. Or we could also
write that the length of segment AC over
the hypotenuse, over this triangle's
hypotenuse, over AB, is equal to DF/DE-- once
again, this green side over the orange side. These are similar triangles. They're corresponding
to each other. So this is equal to DF/DE. And we could keep going, but
I'll just do another one. Or we could say that the
ratio of this side right over here-- this blue
side to the green side of this triangle--
the length of BC/CA is going to be the same as
the ratio between these two corresponding sides, the
blue over the green, EF/DF. And we got all of
this from the fact that these are
similar triangles. So this is true for
any right triangle that has an angle theta. Then those two triangles
are going to be similar, and all of these ratios
are going to be the same. Well, maybe we can give
names to these ratios relative to the angle theta. So from angle theta's
point of view-- I'll write theta
right over here, or we can just
remember that-- what is the ratio of these two sides? Well from theta's point
of view, that blue side is the opposite side. It's opposite-- so the opposite
side of the right triangle. And then the orange side we've
already labeled the hypotenuse. So from theta's
point of view, this is the opposite side
over the hypotenuse. And I keep stating from
theta's point of view because that wouldn't be the
case for this other angle, for angle B. From angle
B's point of view, this is the adjacent
side over the hypotenuse. And we'll think about that
relationship later on. But let's just all think
of it from theta's point of view right over here. So from theta's point
of view, what is this? Well theta's right over here. Clearly AB and DE are still
the hypotenuses-- hypoteni. I don't know how to say
that in plural again. And what is AC, and what are DF? Well, these are adjacent to it. They're one of
the two sides that make up this angle that
is not the hypotenuse. So this we can view as
the ratio, in either of these triangles,
between the adjacent side-- so this is relative. Once again, this is
opposite angle B, but we're only
thinking about angle A right here, or the
angle that measures theta, or angle D right over
here-- relative to angle A, AC is adjacent. Relative to angle
D, DF is adjacent. So this ratio right over
here is the adjacent over the hypotenuse. And it's going to be the same
for any right triangle that has an angle theta in it. And then finally,
this over here, this is going to be
the opposite side. Once again, this was the
opposite side over here. This ratio for
either right triangle is going to be the opposite
side over the adjacent side. And I really want to
stress the importance-- and we're going to do many,
many more examples of this to make this very concrete--
but for any right triangle that has an angle theta, the
ratio between its opposite side and its hypotenuse is
going to be the same. That comes out of
similar triangles. We've just explored that. The ratio between
the adjacent side to that angle that is
theta and the hypotenuse is going to be the same,
for any of these triangles, as long as it has that
angle theta in it. And the ratio,
relative to the angle theta, between the opposite
side and the adjacent side, between the blue side
and the green side, is always going to be the same. These are similar triangles. So given that,
mathematicians decided to give these things names. Relative to the angle
theta, this ratio is always going to be
the same, so the opposite over hypotenuse, they call this
the sine of the angle theta. Let me do this in a new
color-- by definition-- and we're going to extend this
definition in the future-- this is sine of theta. This right over
here, by definition, is the cosine of theta. And this right over
here, by definition, is the tangent of theta. And a mnemonic that will help
you remember this-- and these really are just definitions. People realized, wow, by similar
triangles, for any angle theta, this ratio is always
going to be the same. Because of similar triangles,
for any angle theta, this ratio is always
going to be the same. This ratio is always
going to the same. So let's make these definitions. And to help us
remember it, there's the mnemonic soh-cah-toa. So I'll write it like this. soh is sine is opposite
over hypotenuse. cah-- cosine is adjacent
over hypotenuse. And then finally,
tangent is opposite over adjacent-- soh-cah-toa. And in future videos,
we'll actually apply these definitions for
these trigonometric functions.