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## Trigonometry

### Course: Trigonometry > Unit 1

Lesson 5: Sine and cosine of complementary angles- Intro to the Pythagorean trig identity
- Sine & cosine of complementary angles
- Using complementary angles
- Relate ratios in right triangles
- Trig word problem: complementary angles
- Trig challenge problem: trig values & side ratios
- Trig ratios of special triangles

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# Intro to the Pythagorean trig identity

Sal introduces and proves the identity (sinθ)^2+(cosθ)^2=1, which arises from the Pythagorean theorem! Created by Sal Khan.

## Want to join the conversation?

- Are arccos and secant the same?(16 votes)
- To add to Steven's excellent answer, let note a point at which beginning student get confused.

The inverses of the trig function are indicated either with arc- or with ⁻¹. However, the inverses are NOT the reciprocal of the trig functions. The confusion come from the fact that ⁻¹ is sometimes used to indicate a reciprocal. So, the problem is an inconsistency of notation that has, unfortunately, become standard.

Thus, cos⁻¹ x = arccos x. It does NOT equal sec x or 1 / cos x

The same goes for all of the trigonometric functions and their inverses.

To make it more confusing (cos x)⁻¹ does mean sec x. So, the ⁻¹ only means the inverse function instead of the reciprocal when it is written immediately after the name of the function but before the argument.(48 votes)

- At2:08, he says "sin^2(theta)" why didn't he say "sin(theta)^2"?(22 votes)
- sin ^2 theta is the more "formal" way to write out the equation, as it avoids the possibility of mistaking the theta as "theta^2".(26 votes)

- So I'm doing trig in foundations 11 and we are just reviewing grade 10. In grade 10 I completely understood but now I find it harder than ever and I looked to you for help but I don't know what "THEDA" is ?(7 votes)
- Theta is a Greek letter. It usually represents an unknown angle measure. So...I can literally say...I don't know what Theta is either. Oh I'm gonna die laughing. That was good right? Kidding. Yeah, Theta represents the unknown angle measure, it's like a variable. :)(41 votes)

- Sal writes sin²Θ + cos²Θ = 1, shouldn't it be (sinΘ)² + (cosΘ)² = 1?(10 votes)
- Those mean the same thing. However, it is customary to write it as sin² Θ instead of (sin Θ )² to avoid confusion with sin (Θ²).(20 votes)

- Does the theta have to always bee in that exact place or can it be any unknown angle?(5 votes)
- It can be any angle. Theta just means
*the angle we are evaluating*(9 votes)

- At3:16, it says sin^2 θ+cos^2 θ=1.

So does it mean that sin θ=(1-cos^2 θ)^1/2 and cos θ=(1-sin^2 θ)^1/2?(5 votes)- At3:16right? yes, it is true that sin θ=(1-cos^2 θ)^1/2 and cos θ=(1-sin^2 θ)^1/2, we can achieve that by simple rearrangement of the equation.(3 votes)

- Does this identity apply to non-right triangles as well?(4 votes)
- No, it doesn't. This identity uses a particular property with right triangles called the Pythagorean theorem, where the hypotenuse's length is equal to the square root of the sum of the squares of the legs of the triangle. Plus, the trigonometric ratios themselves apply to right triangles. There are things that apply to all triangles, but I don't want to explain them in this reply because it may be confusing.(1 vote)

- oooooooooooh thanks for the info i always thought about why it equals to 1(4 votes)
- Is sin^2(theta) equal to the sin(theta), and then all of that squared? Or is it theta squared, and then the sine of that? I'm guessing it's the former, but just making sure. . .(3 votes)
- no SIn^2(theta ) is sin squared and then multiplied by theta(0 votes)

- At the near end of the video, Sal says that this identity is part of the motivation for discovering the unit circle definitions. Why? I don't see a connection.(2 votes)
- According to the Pythagorean theorem, the sum of the squares of the lengths of these two sides should equal the square of the length of the hypotenuse:

x² + y² = 1²

But because x = cosθ and y = sinθ for a point (x, y) on the unit circle, this becomes:

(cosθ)² + (sinθ)² = 1

or

cos²θ + sin²θ = 1

So the identity sin²θ + cos²θ = 1 is inherent in the definition of the sine and cosine functions in terms of the unit circle, and it provides a mathematical confirmation of the geometric relationship between the unit circle and right triangles(4 votes)

## Video transcript

So we've got a right
triangle drawn over here where this base's length
is a, the height here is b, and the length of
the hypotenuse is c. And we already know when
we see something like this, we know from the
Pythagorean theorem, the relationship
between a, b, and c, we know there's a
squared plus b squared is going to be equal to
the hypotenuse squared, is going to be
equal to c squared. What I want to do
in this video is explore how we can relate trig
functions to, essentially, the Pythagorean theorem. And to do that, let's pick
one of these non-right angles. So let's pick this angle
right over here as theta, and let's just think about
this what the sine of theta is and what the cosine
of theta is, and see if we can mess with them a
little bit to somehow leverage the Pythagorean theorem. So before we do that, let's
just write down sohcahtoa just so we remember the definitions
of these trig functions. So sine is opposite
over hypotenuse. Cah, cosine is adjacent
over hypotenuse. And toa, tan is
opposite over adjacent, we won't be using tan,
at least in this video. So let's think
about sine of theta. I will do it, I'll do
it in this blue color. So sine of theta is what? It is opposite over
hypotenuse, so it is equal to the
length of b or it is equal to b-- b
is the length-- b over the length of the
hypotenuse, which is c. Now what is cosine of theta? Well, the adjacent
side, the side of this angle that is not the
hypotenuse, it has length a. So it's the length
of the adjacent side over the length
of the hypotenuse. Now how could I
relate these things? Well it seems like, if
I square sine of theta, then I'm going to
have sine squared theta is equal to b
squared over c squared, and cosine squared
theta is going to be a squared over c squared. Seems like I might be able
to add them to get something that's pretty close to the
Pythagorean theorem here. So let's try that out. So sine squared theta is equal
to b squared over c squared. I just squared both sides. Cosine squared theta is equal
to a squared over c squared. So what's this sum? What's sine squared theta
plus cosine squared theta? Is going to be equal to what? Sine squared theta is b
squared over c squared, plus a squared over
c squared, which is going to be equal
to-- Well we have a common denominator
of c squared. And the numerator, we have
b squared plus a squared. Now, what is b squared
plus a squared? Well, we have it
right over here, Pythagorean theorem tells
us, b squared plus a squared or a squared plus b squared is
going to be equal to c squared. So this numerator
simplifies to c squared. And the whole expression is
c squared over c squared, which is just equal to 1. So using the sohcahtoa
definition, in a future video, we'll use the unit
circle definition. But you see just
using the units, just even using the sohcahtoa
definition of our trig functions, we see probably
the most important of all the trig identities. That the sine squared theta,
sine squared of an angle, plus the cosine squared
of that same angle-- I'm introducing
orange unnecessarily-- is going to be equal to 1. Now you might probably
be saying, OK Sal, that's kind of cool, but
what's the big deal about this? Why should I care about this? Well, the big deal is now you
give me the sine of an angle and I can solve this equation
for the cosine of that angle, or vice versa. So this is actually a pretty
powerful, powerful thing. And this is also part
of the motivation even for the unit circle
definition of trig functions.