>> Hi, this is Julie Harland
and I'm Your Math Gal.
Please visit my website
at yourmathgal.com
where you could search for any
of my videos organized
by topic.
>> All right,
here's a Trig problem we're
going to solve.
If the cosine
of 2A equals one-third
and 360 degrees is less
than 2A which is less
than 450 degrees,
this is saying something
about what Quadrant 2A is in.
Find the cosine of A,
the sine of A and the tangent
of A. So what we're given is
the cosine of 2A
and we're trying
to find the cosine
of A. First off,
no you can't just divide by 2
and say it's one-sixth.
These are two totally
different angles.
So what we can do is --
think, well
if we know the cosine of 2A,
do we have a formula
that we've got the cosine
of 2A in?
Well how about the formula
for the cosine of 2A?
The formula for the cosine
of 2A, there are different
ones, but one
of them is 2 cosine squared A
minus 1.
That's one formula.
So if we know the cosine
of 2A is one-third,
we could replace cosine of 2A
with one-third
and we have one-third equals 2
cosine squared A minus 1,
and we're trying
to find the cosine of A.
So that's why I use this
formula, right?
So if I could solve this
equation for the cosine of A,
I'll be done.
Now from here,
there's different ways you can
solve it.
I'm going to do it
by multiplying both sides
by the least common multiple
which is 3 to get rid
of the fractions, okay?
So I'm just multiplying both
sides of the equation 3.
So 3 times one-third is 1,
3 times --
and I'm going
to have the equal,
3 times 2 cosine squared A is,
6 cosine squared A,
and 3 times 1 is 3.
And then we have to add 3
to both sides
so we have 4 equals 6cosine
squared A,
got a little extra mark there.
And then we get
to divide both sides by 6
and so we have the two-thirds
equals cosine squared A.
So I'm going to write
that as cosine squared A
equals two-thirds
and now here's
where the trick comes in.
We'll take the square roots
of both sides, right?
But we have to decide
when you take the square root,
usually you put plus or minus.
But we know the cosine
of this particular angle A
in this problem can't be two
different answers.
So we go back up --
what other information do
we have?
Well we've got
that 360 degrees is less
than 2A is less than 450.
So if we look at this,
we could divide all sides by 2
to find out where A is.
So 360 divided by 2 is 180.
[Background music] 2A divided
by 2 is A, and 450 divided
by 2 is 225.
So this tells you something
about where A is and that's
in Quadrant 3
and in Quadrant 3,
the cosine is negative.
All right,
so we know we're going
to choose the negative square
root here.
So when I take the square root
of both sides,
I need to take the square root
of two-thirds, but it's going
to be the negative square root
of two-thirds,
not positive the square root
of two-thirds.
So that would give me the
square root,
a 2 over the square root of 3
and then I'm going
to just multiple
and denominator
by square roots of 3
because we want
to rationalize
our denominator.
So the cosine
of A is negative square root
of 6 over 3.
And that's how we can get the
cosine of A. Now we could go
and use the other formula
for the cosine of 2A
and do the whole thing again
to figure out what the sine
of A. So one approach would be
you would write the formula
cosine of 2A is 1 minus
[background writing equation]
2 sine squared A,
and we could solve
for the sine
of A doing the same thing.
We already found out that A is
in Quad 3 so it's going
to be negative.
So one way is we can go
through this whole process,
do exactly the same thing,
put in one-third,
multiply everything by 3...
--
[ Background noise ]
...and solve this.
I'm just going
to bring the sine squared
to the left
so it'll be 6 sine squared A
equals 2, and then divide both
sides by 6 to get
to the sine squared A is
one-third and then
when I take the square root
again, because A is
in Quadrant 3,
I'm going to take negative the
square root.
So I'm going
to get the square root of 1
which is just going to be 1
over the square root of 3,
multiply the square
of 3 over 3.
So the sine is negative square
root of 3.
So, by [background music]
using this formula I was able
to find the sine
of A. Now you didn't actually
have to go
through all this work.
You could have used the
information you got
about the cosine here.
So we have the cosine was
negative square root of 6
over 3, right?
And in fact,
we had the cosine squared
of A, it was two-thirds.
We could have used the
Pythagorean Theorem,
I'm sorry --
the Pythagorean Identity,
co squared A,
[background writing equation]
plus sine squared A,
equals 1to find the sine
of A as well.
In other words,
the cosine squared A,
well its right here,
two-thirds,
plus sine squared A equals 1.
Subtract two-thirds
from both sides
and take the square root
and we would end
up with exactly the same
answer, the square --
we would take negative
because it's in Quad 3.
Remember, A is in Quad 3,
so the sine of A is negative.
That's why I'm going
to use this negative
sign here.
I don't know what
happened there.
I just seemed
to accidently get rid of that.
So I'm going
to get the square root of 1
over the square root
of 3 again, we're going
to get negative square root
of 3.
So you really have two ways
to figure out what the sine
of A. I'd say this is easier
but if you don't think
of that, you might just go
to this step here,
use the formula for cosine
of 2A and figure
out what the sine is.
You also could have found
out the sine first
and then figured
out the cosine.
You also could have drawn a
picture to figure it out.
You could have said, "Well,
I'm in Quad 3"
and remember our first thing
we found out,
the cosine
of A was negative square --
so it's 6 over 3, all right?
And the cosine could be
thought of as the X value
over the radius.
So if this is the radius of 3
and this is negative squared
of 6, you could just use the
Pythagorean Theorem to figure
out this length here for Y
to get the cosine.
So Y squared plus a negative
squared --
its 6 squared will just be a
positive 6.
So we'd have
that the Y value is plus
or minus squared of 3.
In this case, again,
you could see the Y is going
down to the third Quadrant.
So it's the negative squared
of 3.
And so the sine of A
from this picture, right,
I can now say
that this is actually negative
squared of 3 would be the Y
value over the radius.
So it's negative squared of 3.
Looking at the picture,
we could also get the tangent.
The tangent is the Y
over the X.
So I would have negative
squared of 3
over negative squared of 6
which is squared 3
over squared 6 or square root
of one-half.
I'm just going to multiply
by 2 over 2 inside
to get square roots
of 2, over 2.
So here's the tangent,
here's the sine
and here's the cosine.
So once you get one of them,
you could draw the picture.
That's one way
of doing it [background
writing equation]
or you could use the formula,
right, to get the sine of it,
or you could have used the
Pythagorean Theorem --
I'm sorry,
[chuckle] the Pythagorean
Identity here, the main one
to get this negative square
roots of 3.
All right,
so there's an example
of how you --
one way of working
this problem.
You could also use half
angle identities.
There's other ways to come
up with the same answer.
But be careful of your signs,
et cetera.
>> Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.
Please visit my website
at yourmathgal.com
where you can view all
of my videos
which are organized by topic.