Video: Trigonometry: An Angle between a Line and the 𝑥-Axis

Video Transcript

In this video, we’re going to see how to apply trigonometry, or more specifically the tangent ratio, to calculate the angle between a line and the 𝑥-axis of a coordinate grid.

First of all, just a very quick reminder of what the tangent ratio is. So we have here a diagram of a right-angled triangle. And I’ve chosen one of the other angles to be labelled as 𝜃. I’ve labelled the three sides with their names — hypotenuse, opposite, and adjacent — according to their position in relation to that angle 𝜃. The tangent ratio for this angle 𝜃, remember, is the ratio of the opposite and the adjacent sides. So it’s defined as the opposite divided by the adjacent.

We can also write down this relationship in terms of the inverse tangent function. So we have that 𝜃 is equal to tan inverse of opposite divided by adjacent. And what this means is if I know what the value of this ratio is, then I can work backwards using the inverse tan function in order to find the value of this angle 𝜃. So in this video, we’re gonna see how to apply the tangent ratio in order to calculate the angle between a line and the 𝑥-axis.

So here is our first question. We’re given a sketch of a coordinate grid. And we have a line segment here joining the point one, zero to the point four, five. And we’re asked to calculate the angle 𝜃, which is the angle made between this line segment and the 𝑥-axis.

So we said we were going to do this using trigonometry, which means we need a right-angled triangle. so I’m gonna sketch in first of all the right-angled triangle beneath this line. So by sketching in that vertical line there, I can see the right-angled triangle that I’m going to use.

Now I can also work out some lengths in this question. That blue line is a vertical line and it goes from 𝑦-coordinate of zero to a 𝑦-coordinate of five. So the length of this line here is five units. I can also work out the length of the base of the triangle, the horizontal side, because it goes from an 𝑥-coordinate of one to an 𝑥-coordinate of four. So it has a length of three units. Now I’m also gonna label the three sides of this triangle as the hypotenuse, the opposite, and the adjacent. That’s always one of the first steps when doing a problem involving trigonometry. So here are the labels for those two three sides.

Now let’s recall that tangent ratio that we talked about previously. And remember it was this: that the tangent of the angle 𝜃 is equal to the opposite divided by the adjacent. So what I’m gonna do is I’m gonna write this ratio out again. But I’m gonna include the information that I know. So I’m gonna replace the opposite with five and I’m gonna replace the adjacent with three because those are their values in this particular question.

So I have that tan of this unknown angle 𝜃 is equal to five over three. In order to work out this angle 𝜃 then, I need to use the inverse tangent function. So this tells me that 𝜃 is equal to inverse tan of five over three. Now at this point, I need to use my calculator. And as the question asked for 𝜃 in degrees, I need to make sure that my calculator is in degree mode. And I’ll type this into my calculator in order to evaluate 𝜃.

So this tells me that 𝜃 is this decimal value here, 59.036 and so on. But the question asked for 𝜃 to the nearest degree, so I need to round my answer. So then I have that 𝜃 is equal to 59 degrees. So in this question, we identified the right-angled triangle that we needed in order to apply the tangent ratio. And then we used the inverse tangent in order to calculate the angle we were looking for.

Okay, the second question we’re going to look at is quite similar. We have a coordinate grid and we have a line segment. And we’re asked to find the angle 𝜃, which you can see labelled on the diagram here. Now we’ll refer to it in a similar way. But we do need to be slightly careful here because this angle 𝜃 is an obtuse angle, whereas previously the angle we used was an acute angle. So we need to think about how that will affect the method that we’re going to use.

So as before, let’s draw in a vertical line down from this point — negative six, three — in order to create the right-angled triangle. So here is that triangle. And as before, we can see some lengths in this triangle. This vertical height here, well it goes from zero to three so this must be three units. And the horizontal side goes from negative six to negative two. So this side here is four units.

Now 𝜃, the angle we’re looking for, isn’t actually in this triangle. But I’m gonna use this angle here. So I’m gonna give this angle a letter. I’m going to call it 𝛼. And what you’ll notice is that 𝛼 and 𝜃 are on a straight line together, which means the sum of these two angles must be 180 degrees. So my method is gonna be to use trigonometry to calculate 𝛼 and then subtract 𝛼 from 180 in order to work out this angle 𝜃.

So let’s start off by labelling the three sides of this triangle: the opposite, the adjacent, and the hypotenuse. And so there are all their labels in relation to this angle 𝛼. Now I need to recall the definition of the tangent ratio again. So I’ve written it down again, but with 𝛼 as the angle rather than 𝜃 in order to save confusion as 𝛼 is the label given to the angle inside the triangle. So I’m gonna write this ratio out. But I’m gonna replace the opposite and the adjacent with their values in this triangle. So that’s three and four.

So I have that tan 𝛼 is equal to three divided by four. So in order to work out 𝛼, I need to use the inverse tan function, which means I have that 𝛼 is equal to the inverse tan of three over four. Now I need to use my calculator to evaluate this. And I have that 𝛼 is equal to 36.86989 and so on.

Now remember, I’m not actually asked to find 𝛼, I’m asked to find 𝜃. So I need to subtract this value from 180 because they’ll sit on a straight line together. So I’m gonna keep this value for 𝛼 on my calculator and just do 180 minus that answer in order to make sure my calculation is as exact as it can be.

So I have 𝜃 is equal to 180 minus this value of 36.869 and so on. So then 𝜃 is equal to 143.13. The question asked me to give 𝜃 to the nearest degree. So I need to round my answer. So my final answer then is that 𝜃 is equal to 143 degrees. And of course, that fits with 𝜃 being an obtuse angle.

So in summary then, in this video, we’ve seen an application of the tangent ratio to calculating the angle that exists between a line segment and the 𝑥-axis.