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The maximum value of the
expression 1 over sine squared of theta plus 3
sine theta cosine theta plus 5 cosine
squared of theta is? So let's rewrite this. So this is 1 over, so I have
a sine squared of theta. And then, at least
in my brain, whenever I see a sine squared
of theta, I always look for a cosine
squared of theta. Because I know that when I take
the sum of them it equals 1. And I don't have just one
cosine squared theta here, I have five cosine
squared theta. So let me just take one of them. So I have a plus
cosine squared theta. And since I took one of them,
I only have four of these left. So plus 4 cosine squared theta. And then I have this stuff. Plus 3 sine of theta
cosine of theta. So what this first
step allowed me to do is just turn these two
characters right over here, sine squared theta
plus cosine squared theta, that is equal to 1. So we've simplified it to 1
over 1 plus-- now let's think about how we can write
cosine squared of theta. I'll write our identities here. Cosine squared of theta. And we've prove these in
the trigonometry playlist. This is equal to 1 plus cosine
of 2 theta all of that over 2. And my goal here is, I really
just want to get everything-- well, I really just
want to simplify it. And maybe we'll do calculus. We'll use a little calculus
to find the minimum value of the denominator,
which would give us the maximum value
of the numerator. So let's see, cosine squared
of theta is equal to this. So 4 times this is just going
to be-- so 4 times this. The 4 divided by 2 is 2. So it's going to be 2
times this numerator. So it's going to be 2
plus 2 cosine of 2 theta. That's this term right here. And then this term
right over here, we could use the trig identity. That sine of two theta
is equal to 2 sine of theta cosine of theta. Or you divide both sides by
2, you get 1/2 sine of 2 theta is equal to sine of
theta cosine of theta. So this right over here,
this part right over here is going to be 1/2
sine of 2 theta. But we're multiplying it by 3. So it's going to be plus
3/2 sine of 2 theta. And let's see, this part
over here clearly simplifies. This is 3. So let me just rewrite it. This is 1 over 3 plus
2 cosine of 2 theta plus 3/2 sine of 2 theta. And we're really just
looking for the minimum. We're looking for the minimum
value of the denominator, which would give us the same as the
maximum value of the numerator. It would just be 1 over
this minimum value. So let's see how low we can
get, assuming we're above 0. Let's see how low we can get
for this denominator right here. We're just going to look
for its minimum value. So one thing we can do,
just to simplify things, the minimum value of this
is going to be the same. The min of this thing-- I
don't want to write it there because it kind of
confuses the problem. The minimum value of 3 plus 2
cosine 2 theta plus 3/2 sine of 2 theta is going to be the
same thing as the minimum value of 3 plus-- I'm just going
to do the substitution that 2 theta is equal to x. That just simplifies
things a little bit. You don't have to do that. But just to, so 3 plus 2
cosine of x plus 3/2 sine of x. So this is a pretty
simple expression. Let's see how we can figure
out its minimum value. And my temptation is to take
the derivative, find out where the derivative
is equal to 0, and then that will either be
a maximum or a minimum point. So let's take the derivative. The derivative of
this expression right over here
with respect to x. Well, derivative of 3
with respect to x is 0. Derivative of cosine of x
is negative 2 sine of x. Derivative of 3/2
sine of x is going to be plus 3/2 cosine of x. And then, that is
going to be equal to 0. We want to find where the
slope is 0 because it's either going to be a maximum
or minimum point. And let's see, we
can add, we could add 2 sine of x to both sides. So we get 3/2 cosine of x
is equal to 2 sine of x. And then, we can divide both
sides of this equation by, let's divide it by 2 first. I don't want to
skip too many steps. So 3/4 cosine of x is
equal to sine of x. Then we could divide both
sides by cosine of x. So we get 3 over
4 is equal to sine of x over cosine of x, which is
the same thing as the tangent of x. So an x value that gives us--
or the tangent of an x value that gives us 3/4
is going to give us either a maximum
or a minimum point. So let's think about this. Let's think about
this a little bit. Let me draw my unit circle. So let's think about
two x values that will give us the tangent of 3/4. So let me draw my unit circle. That's a unit circle. Let me draw a unit. This is always the hardest part. So let me draw this. All right, there
is my unit circle. So how can I get a triangle,
or, well, yeah, let's think about that way. How can I get a triangle where
an angle is the tangent of 3/4. Remember, tangent is
opposite over adjacent. Right? Tangent is opposite
over adjacent. So if this is my triangle
right over here, if this is x, opposite over adjacent
is equal to 3/4. So opposite could be 3
and adjacent could be 4. And we hopefully
immediately recognize this. This is a 3, 4, 5 triangle
because it's a right triangle. 3 squared plus 4 squared
is 25 which is 5 squared. So this is a 3, 4, 5 triangle. Now, there's two tangent values. So x could be like this. So this obviously isn't a
unit hypotenuse over here. But we could divide everything
by 5, and it would be. So we could have this situation. We could have this
situation over here. Where this is x. If this is the unit circle,
the hypotenuse is 1. This is 3/5 and this is 4/5. This would, tangent of x here,
tangent of x would give us 3/4. But is this going to give us a
maximum value or minimum value? Well over here, both
cosine of x and sine of x are going to be positive. So both of these are going
to be positive values. So it's going to, probably,
maximize this expression over here. Now the other x that gives us
the same tangent, and remember, the tangent is really just the
slope of the radius in the unit circle, would be this angle. This has the same tangent value. So this x over here. In this case, the tangent
is still going to be 3/4. But over here, the sine
and cosine are negative. So over here, the
x-coordinate, or the cosine, is going to be negative 4/5. And the sine value,
or the y value, is going to be negative 3/5. And this will give
us a minimum point, because here both the sine
and the cosine are negative. So let's use this
x right over here. And notice, we don't even have
to figure out what the x is. Because we know that if the
tangent over here is 4/5, either both the sine
is going to be 3/5, and the cosine is
going to be 4/5, which will give us
a maximum point. Or the tangent could be 3/4. And then the sine
will be negative 3/5, and the cosine will
be negative 4/5. So let's use these over here. So the minimum is
going to be equal to 3 plus 2 times cosine of x. We're using this one over here. So 2 times cosine of x. Cosine of x here
is negative 4/5. And then plus 3/2
times the sine of x. Sine of x here is negative 3/5. And what is this
going to be equal to? This is going to be equal to 3
plus, this is negative 8/5, 3, maybe I should write 3
minus 8/5 minus 9/10. And so this is going
to be equal to-- we could put everything over 10
--30 over 10 minus 16 over 10, that's 8/5. Minus 9/10. And this gives us what? This gives us 5/10. 5/10 or 1/2. So the minimum value for
our denominator-- everything we've been dealing with so
far has been our denominator-- the minimum value of all
of this business over here, the minimum value is 1/2. So the maximum value, that
this whole expression takes, is when the minimum
value is 1/2. So we get 1 over 1/2
half which is equal to 2. And we're done.