Main content

## Sine and cosine of complementary angles

# Intro to the Pythagorean trig identity

CCSS.Math:

## 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.