Fun Trig Problem

A trig problem involving the quadratic equation.

I received a problem
from Bradley.

I don't know his last name.

I'm assuming it's a he.

I don't know where he lives.

But the problem he
gave is interesting.

And I don't think I've
covered this before.

So I think it's worth covering.

So the problem he gave, if I
read his note properly, is

this: 3 sine squared of x is
equal to 1 plus cosine of x.

So at first cut, this seems
like a difficult problem.

How do I-- You
can't solve for x.

You would have arcsines
and the square roots

and cosines et cetera.

Et cetera.

So the way I approached this
is-- Any time that if I see a

cosine x here but then I see a
sine squared x here, I start

thinking of what trig
identities are at my disposal.

And what trig identities
involve a sine squared x?

Well the most basic trig
identity, and this comes out

of the unit circle definition
of trig functions, is that

sine squared x plus cosine
squared x is equal to 1.

And that comes out of the fact
that the equation of a circle

is x squared plus y squared is
equal to the radius squared.

But it's the unit circle.

It's equal to 1 squared.

But anyway.

Hopefully you have this
memorized if you've already

been watching the trig videos.

So what does sine
squared x equal?

Well let's solve for it.

So sine squared x is equal to 1
minus cosine squared x, right?

So we could substitute this
term right here with this.

And what does that get us?

Well we're just playing around
at this point, but at least

that way, everything is
in terms of cosine of x.

So let's do that.

Let's substitute.

So we get 3 times
sine squared of x.

We just showed that sine
squared of x is the same thing

as 1 minus cosine squared of x.

Is equal to 1 plus cosine of x.

We can simplify a little bit.

3 minus 3 cosine squared of x
is equal to 1 plus cosine of x.

I don't know.

Just for kicks, let's put
everything onto the right

side of the equation.

And you'll see it
wasn't just for kicks.

0-- right?

I'm just going to --is equal
to-- let's put this onto

the right side --3
cosine squared x.

And then-- Let's see.

We have to subtract
3 from this side.

Well let's just
write the cosine x.

Plus cosine x.

And then 1 minus 3 is minus 2.

Let me make sure I didn't
make a careless mistake.

We have negative 3 here.

We added 3 cosine x--
3 cosine squared of x

to both sides, right?

We subtracted 3
from both sides.

Minus 2 and this cosine
x is this cosine x.

Now what can we do?

Well this is where it
gets interesting.

Because this looks an awful lot
like a quadratic equation

except for the fact that
instead of having ax squared

plus bx plus c, we have
a cosine squared x.

So instead of just having an
x squared, we have a whole

cosine of x squared.

So what do I mean by that?

Let me make a substitution.

And then I think it'll
all become clear to you.

Let's make the substitution
that a-- and I'm just picking

the letter a arbitrarily
--is equal to cosine of x.

So if we were to take the
cosine x's of this and replace

them with a, what do we get?

And I'm just going to
switch it around.

So I want to put the
0 on that side.

Equal 0.

So we get 3-- Well
cosine squared x.

That's the same thing as
cosine of x squared, right?

So we get 3a squared plus
a minus 2 is equal to 0.

Well now we have a
pure quadratic.

And we can solve it using
the quadratic equation.

So what's the
quadratic equation?

Let me write it up here.

Negative b plus or minus
the square root of b

squared minus 4ac.

All of that over 2a.

So what are the roots
of this equation?

Well what's minus-- And
remember this a is

different than this a.

Maybe I shouldn't have
used a as a letter.

But these-- a, b, and c in
the quadratic equation

represent the coefficients.

So this is a.

b is 1.

And c is just minus 2.

So what are the roots of this?

So the a's that solve
this. a can equal--

And I know I confused you.

I could-- Let me actually
write it different.

Let's make this, instead of
a is equal to cosine x, let

me say that-- I don't know.

Let me pick a good letter
that's not involved in

the-- Let me say d.

So 3d squared plus
d minus is 2.

So now the a's, b's and c's are
definitely the coefficients.

So the solutions to this are
d-- because I didn't want to

use a, b, or c --d is
equal to minus b.

Well, b is 1.

Minus 1.

And if this is completely
foreign to you, you should

review the videos on
the quadratic equation.

Minus b squared.

Well that's 1 squared.

Minus 4ac.

So minus 4 times a,
times 3, times c.

Well c is minus 2, right?

So we get a-- That
minus cancels there.

And we have a 2 there.

All of that over 2 times a. a
is 3, so we have it over 6.

So this equals minus 1
plus or minus the square

root-- What is this?

4 times 3 times 2.

24 plus 1.

25.

Oh.

This works out cleanly.

Over 6.

So that equals minus 1
plus or minus 5 over 6.

And so what are the roots?

The roots are-- What's
minus 1 minus 5?

That's minus 6 over 6.

So it's minus 1.

What's the other one?

Minus 1 plus 5 is 4.

4 over 6 is 2/3.

So the solution is to
the equation-- Let me

clear up some space.

Hopefully it'll let me
clear up some space here.

Let me see.

What was I doing?

Oh.

Maybe I want to leave--
I can get rid of this.

You know the identity.

And you also know the
quadratic formula.

Let's see.

Actually, let me get
rid of this too.

Clear up a bunch of space.

I wanted to leave this here
because this showed how this

turned into a quadratic, but
instead of having it in terms

of just a variable, we have
it in terms of cosine of x.

And then we made this d
is equal to cosine of x.

Anyway.

So the solution to this
equation is that quadratic.

Is d is equal to minus
1 or 2/3, right?

But, of course, we made the
substitution long ago that

d is equal to cosine of x.

So the solution to this
equation, in terms of x, is the

solution to this equation.

Cosine of x is equal to minus 1
or cosine of x is equal to 2/3.

Well this one's easy, right?

x is equal to
arccosine of minus 1.

I always forget if there's two
c's when you do arccosine.

Anyway, so what-- At what
degree or radian value does the

cosine of x equal minus 1?

Well it's at pi, right?

So x could equal pi, which
is also or 180 degrees.

This one is not as easy.

I think I will have to use
a calculator for this.

Unless I'm-- whoops.

So you may not realize it,
but Google is actually

a calculator.

And a far more advanced
calculator than most.

So we could use Google
to figure out the

arccosine of 2/3.

Let's do that.

Arccosine-- and I don't
know if I'm spelling it

right --of 2 over 3.

Google tells us that it's
0.841 and a bunch of numbers.

So x is equal to
arccosine of 2 over 3.

So x is equal to 0.84106.

Let's see if they work.

Let's, just for fun, let's
just see if this one works.

Let's see if we substitute
pi into this equation we

get the correct answer.

Well what's sine of pi?

Let me erase all of this
is so we can check it.

I'm only going to check pi.

The 0.84.

And I don't know.

That's messy.

But you could do that
in your own time.

So let's check pi.

x equals-- No.

That's not what I wanted to do.

So what is-- Let's make
sure this works with pi.

3 sine squared of pi is equal
to 1 plus cosine of pi.

Well what's sine of pi?

This is equal to 3
sine of pi squared.

This is equal to 1
plus cosine of pi.

Well sine of pi is 0, right?

The y-value when you
go 180 degrees is 0.

So this is 0.

And what's cosine of pi?

Cosine of pi is negative 1.

So 1 plus minus 1.

Well this is true.

So pi worked in that equation.

I think if you substitute
that 0841068 whatever, you'd

also find that that works.

So thanks Bradley
for sending this.

I thought this was a neat
problem because it looks

like it's trigonometry.

And it was trigonometry but
you had to know a little

bit of identities.

And then you had to recognize
it as a quadratic equation.

I will see you in
a future video.