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>> This is YourMathGal,
Julie Harland.
Please visit my website
at yourmathgal.com where all
of my videos are organized
by topic.
All right,
here's our next example
for finding the tangent
of a sum.
So we want to find the tangent
of A plus B,
given the triangles below
where we see angle A,
we see angle B. Now, remember,
the sum means
if you took angle A,
which looks
like it's something
like this big,
and then you're adding
on the smaller angle B, and--
so then we'd somehow have
to figure out the tangent
for that angle here.
Okay? So we want
to use the formula.
So, remember the tangent
of A plus B formula is tangent
of A plus tangent of B, well,
over 1 minus the tangent
of A times the tangent of B.
So we need to look
at our pictures to be able
to fill in what the tangent
of A and the tangent of B is,
and then just fill it right
in there.
So let's go ahead and do that.
So what's the tangent of A?
I look at A over here,
the tangent is 3/4.
So I'm going to write 3/4,
that's the tangent of A. plus,
and then I look over here,
what's the tangent of B?
Well, that would be 5/12.
And the denominator is 1
minus, and then you multiply
those two together,
3/4 times 5/12.
Now, that could be simplified,
just keep in mind this
denominator here is really 48.
So to simplify this
as a complex fraction,
you want to multiply numerator
and denominator
by the least common multiple
to get rid
of the complex fraction.
And that's one way to do it,
I just find that the easiest.
So the least common
denominator here--
well, if it was just the
numerator I cared about,
you know, it could be 12.
But that won't work
in the denominator 'cause I
have 4 times 12 right here.
So, the least common
denominator is going to be 48.
So let's see,
I'm going to do a little
space here.
So we could this step.
So I'm going
to do 48 times 3/4
and then 48 times 5/12,
and then same thing down here.
I've got to multiply 48
by each term.
So it's going
to be 48 times 3/4,
plus 48 times 5/12.
The denominator,
I'm going to multiply it also.
48 times 1 plus 48 times--
now, if you want,
you could simplify this
and write 15/48,
or if you want,
you could have left it
as 3/4 times 5/12.
To conserve space,
I just multiplied it
since nothing canceled there.
Right now,
we see what we can cancel,
4 goes into 48 twelve times,
and 12 goes
into 48 four times.
Nothing cancels
with 48 times 1,
but both of these 48
is cancel.
So what does that give me?
Well, in the numerator,
I now have my 12 times 3,
make sure you see those get
multiplied together.
So I'm going to get 36 plus,
and then in here I've got my 4
times my 5, that's 20.
In the-- now,
in the denominator,
I've got to do 48 times 1,
right?
Which is 48.
And this canceled, right?
So I just have plus 15.
Now you can't cancel any
of these.
Word of operations means
you've got
to simplify the numerator
and simplify the denominator.
So, I just add
up the numerators, that's 56
and add up, the numbers
in the denominator is 63.
[ Pause ]
And so, that is the tangent
of A plus B,
if these are our triangles.
And you might
like to take a look again
and see if that seems kind
of reasonable, you know,
like here's A is
about that big and B is
about that big,
and does it seem reasonable
that, like it's almost 1,
yeah, it kind of does to me.
It looks like this side
over here is a little bit less
than that side.
So that's just an
approximation, you know,
when you're free hand drawing
it so it might not always look
exactly right.
But, you know,
it looks like it's
in the ball park.
And so, we're just using our
formula for tangent
of A plus--
I mean, tangent of A plus B,
and looking at the pictures
to get our exact answer.
This is YourMathGal,
Julie Harland.
Please visit my website
at yourmathgal.com where all
of my videos are organized
by topic.