### Video Transcript

In this video, we’re going to look
at the trigonometric ratio tangent. And we’re going to see how it can
be used in order to find the length of missing sides or the size of missing angles
in right-angled triangles. Now, first let’s have a look at
what the tangent ratio is. So you see I’ve drawn a
right-angled triangle. And I’ve labeled one of the two
other angles with the Greek letter 𝜃. Now often, the first step in any
problem involving trigonometry is identifying the names of the three sides in
relation to the angle we’re interested in. So in this case, that’s in relation
to this angle 𝜃.

In a right-angled triangle,
remember, the longest side, the side that’s opposite the right angle, is always
called the hypotenuse. So that is this side here. The other two sides in a
right-angled triangle are called the opposite and the adjacent. And those names depend on their
position relative to this angle 𝜃. So the opposite is the side
opposite the angle 𝜃. It’s the one side that isn’t
involved in making that angle. So for this triangle, it’s this
side here. Finally, the adjacent is the side
between the angle and the right angle. So that’s this third side here.

Now, trigonometry is all about the
different ratios that exist between these pairs of sides for specific values of this
angle 𝜃. You’ve probably already met the two
trigonometric ratios sine and cosine, or sin and cos as they’re often abbreviated
to. In this video, we’re looking at the
tangent ratio, or tan as it’s also known. So the tangent ratio for a
particular angle 𝜃 is the ratio between the opposite and the adjacent sides. And so it’s always calculated by
dividing the length of the opposite side by the length of the adjacent side. If you’re familiar with SOHCAHTOA
in trigonometry, then it’s this TOA part here. The tan of an angle is equal to O,
the opposite, divided by A, the adjacent.

So if we know the length of one of
these two sides and we know the angle, we can use this relationship here in order to
work out the length of the other side. The other thing we might want to do
is work out the size of the angle when we know both the opposite and the
adjacent. And in order to do that, we need
something called the inverse tangent. Now, this is expressed using this
notation here, tan, and then a superscript negative one. And this tells us that the angle 𝜃
is equal to the inverse tan of the opposite over the adjacent. What this means is if I know what
the ratio is between those two pairs of sides, it enables me to work backwards to
work out what the angle is that creates that ratio. So those are the two key facts that
we need to remember throughout this video. We’ll now see how to apply them to
working out some missing lengths and some missing angles in right-angled
triangles. So here is our first problem.

We’re given a diagram of a
right-angled triangle. And we’re asked to find the value
of 𝑥 to the nearest tenth.

Looking at the diagram, we can see
that 𝑥 represents a missing side. And we’re also given the length of
one side and the size of one of these angles in addition to the right angle. So my first step with any
trigonometry problem is always to label up the three sides with their names, the
hypotenuse, the adjacent, and the opposite. So I’ve just used the first letter
of each of those words to label them up. Now, let’s recall that definition
of the tangent ratio. And, remember, it was that tan of
𝜃, the angle, is equal to the opposite divided by the adjacent. So what I’m gonna do is I’m gonna
write down this ratio for this specific triangle. I’m going to replace 𝜃, which is
the angle with 38 degrees. And I’m gonna replace the opposite
with four because that’s its length in the diagram. And I’m gonna replace the adjacent
with 𝑥 because that’s the label that’s been given to it.

So now, I have this statement here,
tan of 38 is equal to four over 𝑥. So this is an equation that I can
solve in order to work out the value of this missing letter 𝑥. So 𝑥 is in the denominator of a
fraction on the right-hand side. So in order to bring it out of the
denominator, I’m going to multiply both sides of this equation by 𝑥. So when I do that, I have 𝑥 tan 38
is equal to four. Now in order to work out what 𝑥
is, the next step is just to divide both sides of this equation by tan 38. Tan 38 is just a number. So I can do that. So this gives me 𝑥 is equal to
four divided by tan 38. Now, we are going to need a
calculator in order to answer this question. Your calculator has the values of
sin, cos, and tan for all of these different angles already programmed into it. So I can type tan 38 into my
calculator. And it will recall that value for
me.

There are some angles — 30 degrees,
45 degrees, 60 degrees — for which the values of these sin, cos, and tan ratios are
actually relatively straightforward. And they can be written down
exactly in terms of surds. So for those angles, it is possible
to do trigonometry without a calculator. But we would do need it in this
example. So in my calculator, I’m gonna type
four divided by tan 38. And when I do that, it tells me
that 𝑥 is equal to 5.119766 and so on, this decimal here. Now, it is worth just pointing out
here that the angle that I was given was measured in degrees. So I needed to make sure that my
calculator was in degree mode. And that’s always the case when
doing trigonometry. You need to make sure your
calculator is in the same mode as the way that the angle is measured in the
question.

Now this question asked me for the
value of 𝑥 to the nearest tenth. So I need to round my answer. So I have that 𝑥 is equal to
5.1. So a reminder of what we did then,
we recalled the definition of that tangent ratio. Then we substituted the values
given in the question to the relevant places and then solved the resulting equation
in order to find the value we were looking for. Okay the next question, we’re given
a diagram of a right-angled triangle. And we’re given two sides this
time. The question asks us to find the
value of 𝜃 to the nearest degree. So this time, we’re being asked to
find an angle rather than a side length.

So my first step is gonna be to
label the three sides of this right-angled triangle with the letters representing
their names. So there they are. You do have to be slightly careful
with this because, remember, triangles can be drawn in lots of different
orientations. The hypotenuse is just always the
side opposite the right angle. But you just have to think
carefully about the adjacent and the opposite depending on which angle has been
labeled. So let’s recall our definition of
the tangent ratio. And here it is. Tan of 𝜃, the angle, is equal to
the opposite divided by the adjacent. So I’m gonna use this ratio. And I’m gonna substitute the known
values of the opposite and the adjacent. So this tells me that tan of this
angle is equal to 5.2 divided by 2.8.

Now as we’re looking to calculate
an angle this time, we need to use the inverse tan function that says given that I
know what the ratio is, I need to work backwards to work out the angle that that
ratio belongs to. So this tells me that the angle 𝜃
is equal to the inverse tan of 5.2 over 2.8. Now, you could perhaps go straight
to that stage of working out if you preferred. So at this stage, I’m gonna need to
use my calculator in order to evaluate this. Now that inverse tan button is
often located directly above the tan button. You often have to press shift to
get to it. But that will depend on the
particular calculator that you have. So evaluating that on my calculator
gives me this decimal value here. But I’m asked to find 𝜃 to the
nearest degree, so I need to round my answer.

So then this tells me that 𝜃 must
be equal to 62 degrees. So in this question, we began in
exactly the same way. But because it was an angle that we
were finding this time as opposed to the length of a side, we needed to use that
inverse tan function. Okay, our final question is a
worded problem. So let’s read it through
carefully.

A ladder leans against a wall
making an angle of 15 degrees with the wall. The base of the ladder is 0.5
meters from the base of the wall. And the question we’re asked is,
how far up the wall does the ladder reach?

So with a worded question like
this, if I’m not given a diagram, I would always draw my own to start off with. So we’re gonna have a diagram of a
ladder, a wall, and a floor. And we’re making the assumption
here that the wall is vertical and the floor is horizontal. That seems a reasonable assumption
for this question. So here is a sketch of the wall,
the floor, and the ladder. Because we assume the wall is
vertical and the floor horizontal, we know that we have a right angle here. Now we need to put on the
information we’re given. So we’re told the ladder makes an
angle of 15 degrees with the wall. So this angle here is 15
degrees. And we’re also told the base of the
ladder is 0.5 meters from the base of the wall. So this measurement here is 0.5
meters.

Now what we’re asked to find is how
far up the wall the ladder reaches. So we’re being asked to find this
measurement here, which I’ll call 𝑦 meters. So I’ve got my diagram. And I can see that actually it’s
just a problem about a right-angled triangle. So we’re gonna approach it in
exactly the same way as the previous ones. I’m gonna start off by labeling the
three sides as always. So I have the hypotenuse, the
adjacent, and the opposite. Now let’s recall that tangent ratio
that we’re going to need in this question. So I have that tan of the angle 𝜃
is equal to the opposite over the adjacent. You’d be becoming familiar with
that by now. So as in the previous questions,
I’m gonna write down this ratio again. But I’m gonna fill in the
information I know.

So I know that the angle 𝜃 is
15. And I know in this case that the
opposite is 0.5. So I have that tan of 15 is equal
to 0.5 over 𝑦. Now I need to solve this equation
in order to work out the value of 𝑦. So 𝑦 is in the denominator of this
fraction. So I’m gonna multiply both sides by
𝑦 in order to bring it up into the top numerator. And when I do that, I have 𝑦 tan
15 is equal to 0.5. Now remember tan 15 is just a
number. So I can divide both sides of the
equation by it. So I’ll have 𝑦 is equal to 0.5
over tan 15. Now this is the stage where I reach
for my calculator in order to evaluate this. And it tells me that 𝑦 is equal to
1.86602 and so on.

Now I need to choose a sensible way
to round that answer cause I haven’t been asked for a specific level of
accuracy. So the other measurement of 0.5
seems to be given to the nearest tenth. I’ll do the same level of rounding
for this value of 𝑦 here. So that will give me that 𝑦 is
equal to 1.9. And to answer the question how far
up the wall does the ladder reach, I put the units back in. It reaches 1.9 meters up this
wall.

So to summarize then, in this
video, we’ve seen the definition of tangent as the ratio between the opposite and
adjacent sides of a right-angled triangle. We’ve seen how to apply both
tangent and inverse tangent to problems involving right-angled triangles in order to
find a missing side or a missing angle. And we’ve seen how to apply this to
a worded problem.