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, the 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. I’m 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.