Using Trig Functions

Using Trigonometric functions to solve the sides of a right triangle

We're now going to do a few
examples to actually show

you why the trig functions
are actually useful.

So let's get started
with a problem.

Let's say I have this
right triangle.

That's my right triangle.

There's the right angle.

And let's say I know that
the measure of this angle

is pi over 4 radians.

And I'll just write
rad for short.

If the measure of this angle is
pi over 4 radians, and I also

know that this side of the
triangle-- this side right

here-- is 10 square roots of 2.

So I know this side
of the triangle.

I know this angle, which
is pi over 4 radians.

And now, the question is, what
is this side of the triangle?

I'm going to highlight that.

And let me make it in orange.

So let's figure out what
we know and what we

need to figure out.

We know the angle,
pi over 4 radians.

And actually, turns out if
you were to convert that

to degrees, it would
be 45 degrees.

And we know-- what
side is this?

This is the hypotenuse
of the triangle, right?

And what are we trying
to figure out?

Are we trying to figure out
the hypotenuse, the adjacent

side to the angle, or the
opposite side to the angle?

Well, this is the hypotenuse,
we already know that.

This is the opposite side.

This is the opposite side.

And this yellow side is
the adjacent side, right?

It's just adjacent
to this angle.

So we know the angle, we know
the hypotenuse, and we want to

figure out the adjacent side.

So let me ask you a question.

What trig function deals
with the adjacent side

and the hypotenuse?

Because we have the adjacent
side is what we want

to figure out, and we
know the hypotenuse.

Well, let's write down
our mnemonic, just in

case you forgot it.

SOHCAHTOA.

So which one uses
adjacent and hypotenuse?

Right?

It's CAH.

And CAH, the c is for what?

The c is for cosine.

Cosine of an angle-- let's just
call it any angle-- is equal to

the adjacent over
the hypotenuse.

So let's use this information
to try to solve for this orange

side, or this yellow side.

So we know that cosine of pi
over 4 radians-- so let's say

cosine of pi over 4-- must
equal this adjacent

side right here.

Let's just call that
a. a for adjacent.

The adjacent side divided
by the hypotenuse.

The hypotenuse is this side.

And in the problem, we
were given that it's

10 square roots of 2.

So we can solve for a by
multiplying both sides of

this equation by 10
square roots of 2.

And we will get--
because, right?

If we just multiply times
10 square root of 2,

these cancel out.

And then you get a 10
square root of 2 here.

So you get a is equal to 10
square roots of 2 times

the cosine of pi over 4.

Now you're probably saying,
Sal, this does not look too

simple, and I don't know how
big the cosine of pi over 4 is.

What do I do?

Well, no one has the trig
functions, or the values of

the trig functions memorized.

There's a couple
of ways to do it.

Either I could give you what
the cosine of pi over 4 is.

That's sometimes
given in a problem.

Or you can make sure that your
calculator is set to radians

and you can just type in pi
divided by 4-- which is roughly

0.79-- and then press
the cosine button.

You finally know
what it's good for.

And you'll get a value.

Or-- and this is kind of the
old school way of doing it--

there are trig tables where
you could look up what cosine

of pi over 4 is in a table.

Since I don't have any of that
at my disposal right now,

I'll just tell you what the
cosine of pi over 4 is.

The cosine of pi over 4 is
square root of 2 over 2.

So a, which is the adjacent
side-- a for adjacent-- is

equal to 10 square roots of 2
times square root of 2 over 2.

Remember, to get the square
root of 2 over 2, you might

be a little confused.

You're like, how
did Sal get that?

All I said is, the cosine
of pi over 4 is square

root of 2 over 2.

And that's not something that--
well, actually, this one you

might know offhand, because
of the 45 degree angle.

But this isn't something
that people memorize.

This is something you would
look up, or it's given in

the problem, or you'd
use a calculator for.

And a calculator, of course,
wouldn't give you square

root of 2 over 2.

It'd give you a decimal
number that's not obviously

square root of 2 over 2.

But anyway, I told you that the
cosine of pi over 4 is the

square root of 2 over 2.

And so if we multiply, what's
the square root of 2 over 2?

What's the square root of 2
times the square root of 2?

It's 2.

So that's 2, and then that
cancels with that 2.

And so everything cancels
except for the 10.

So the adjacent side
is equal to 10.

Let's do another one.

Let me delete this.

Give me 1 second.

I'm actually-- this is one of
the few modules that I'm not

generating the problems on the
fly, because I need to make

sure that I actually have
the trig function values

before I do the problem.

So let's say I have
another right triangle.

I probably shouldn't have
deleted that last one.

So let's see, this is
my right triangle.

How much time do I have--
about 4 minutes left.

Should be enough.

So this is my right triangle.

And I know the angle--
let's call this--.

I know this angle right
here is 0.54 radians.

And I also know that this side
right here is 3 units long.

And I want to figure
out this side.

So what do I know?

Well, this side is what side
relative to the angle?

It's the opposite side, right?

Because the angle is here, and
we go opposite the angle.

So this is the opposite side.

And what's this side?

Is this the adjacent side,
or is it the hypotenuse?

Well, this is the
hypotenuse, right?

The long side, and it's
opposite the right angle.

So this is the adjacent side.

So what trig function uses
opposite and adjacent?

Let's write down
SOHCAHTOA again.

SOHCAHTOA.

TOA uses opposite and adjacent.

OA.

So T for tangent, right?

TOA.

So tangent is equal to
opposite over adjacent.

So let's use that.

So let's take the tangent
of 0.54 radians.

So the tangent of 0.54 will
equal the side opposite to it.

So that's 3, right?

The opposite side is 3.

Over the adjacent side.

Well, once again, the adjacent
side is what we don't know.

So we have to solve for a.

So if we multiply both sides
by a, we get a tan of 0.54--

we could do that because we
know it's not 0-- equals 3.

Or a is equal to 3 divided
by the tangent of 0.54.

So once again, I don't have
memorized what the tangent of

0.54 is, but I will tell you
what it is because you also

don't have it memorized.

Or you could use a calculator
to figure it out if you

had a radian function.

The tangent of 0.54 is equal
to-- let me make sure

I have this right.

Oh, right.

The tangent of 0.54 is 3/5.

So then a is equal
to 3 over 3/5.

Right, the adjacent side--
now, once again, how

did I get this 3/5?

Well, I just told you.

Or you can use a calculator
to know that the

tangent of 0.54 is 3/5.

And of course, I'm using
numbers that work out

well, just so that the
fractions all cancel.

So we know that the adjacent
side is equal to-- when you

divide by fractions, it's like
multiplying by the numerator.

Multiplying by the inverse.

So times 5/3.

So the adjacent side
is equal to 5.

There.

There you go.

So let's just think
about what I always do.

I think about what I have,
what sides I have, and what

side I want to solve for.

And in this case, it was the
opposite side I had, and I

wanted to solve for
the adjacent side.

And I said, what trig function
involves those 2 sides?

The opposite and the adjacent.

I wrote down SOHCAHTOA.

I said, oh, TOA.

Opposite and adjacent.

That's tan.

So I took the tan of the angle.

And then I said, the tan of the
angle is equal to the opposite

side divided by the
adjacent side.

That's right here.

And then I just solved
for the adjacent side.

And of course, I used a
calculator, or I told you

what the tangent of 0.54 is.

I think I'll do a couple more
of these problems in the next

module, but I'm out
of time for now.

Have fun.