### Video Transcript

In this video, weโre going to see how to apply trigonometry, or more specifically the tangent ratio, to calculate the angle between a line and the ๐ฅ-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 Iโve chosen one of the other angles to be labelled as ๐. Iโve labelled the three sides with their names โ hypotenuse, opposite, and adjacent โ according to their position in relation to that angle ๐. The tangent ratio for this angle ๐, remember, is the ratio of the opposite and the adjacent sides. So itโs 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 ๐ 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 ๐. So in this video, weโre gonna see how to apply the tangent ratio in order to calculate the angle between a line and the ๐ฅ-axis.

So here is our first question. Weโre 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 weโre asked to calculate the angle ๐, which is the angle made between this line segment and the ๐ฅ-axis.

So we said we were going to do this using trigonometry, which means we need a right-angled triangle. so Iโm 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 Iโm 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 ๐ฆ-coordinate of zero to a ๐ฆ-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 ๐ฅ-coordinate of one to an ๐ฅ-coordinate of four. So it has a length of three units. Now Iโm also gonna label the three sides of this triangle as the hypotenuse, the opposite, and the adjacent. Thatโs always one of the first steps when doing a problem involving trigonometry. So here are the labels for those two three sides.

Now letโs recall that tangent ratio that we talked about previously. And remember it was this: that the tangent of the angle ๐ is equal to the opposite divided by the adjacent. So what Iโm gonna do is Iโm gonna write this ratio out again. But Iโm gonna include the information that I know. So Iโm gonna replace the opposite with five and Iโm gonna replace the adjacent with three because those are their values in this particular question.

So I have that tan of this unknown angle ๐ is equal to five over three. In order to work out this angle ๐ then, I need to use the inverse tangent function. So this tells me that ๐ 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 ๐ in degrees, I need to make sure that my calculator is in degree mode. And Iโll type this into my calculator in order to evaluate ๐.

So this tells me that ๐ is this decimal value here, 59.036 and so on. But the question asked for ๐ to the nearest degree, so I need to round my answer. So then I have that ๐ 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 weโre going to look at is quite similar. We have a coordinate grid and we have a line segment. And weโre asked to find the angle ๐, which you can see labelled on the diagram here. Now weโll refer to it in a similar way. But we do need to be slightly careful here because this angle ๐ 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 weโre going to use.

So as before, letโs draw in a vertical line down from this point โ negative six, three โ 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 ๐, the angle weโre looking for, isnโt actually in this triangle. But Iโm gonna use this angle here. So Iโm gonna give this angle a letter. Iโm going to call it ๐ผ. And what youโll notice is that ๐ผ and ๐ 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 ๐ผ and then subtract ๐ผ from 180 in order to work out this angle ๐.

So letโs 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 ๐ผ. Now I need to recall the definition of the tangent ratio again. So Iโve written it down again, but with ๐ผ as the angle rather than ๐ in order to save confusion as ๐ผ is the label given to the angle inside the triangle. So Iโm gonna write this ratio out. But Iโm gonna replace the opposite and the adjacent with their values in this triangle. So thatโs three and four.

So I have that tan ๐ผ is equal to three divided by four. So in order to work out ๐ผ, I need to use the inverse tan function, which means I have that ๐ผ is equal to the inverse tan of three over four. Now I need to use my calculator to evaluate this. And I have that ๐ผ is equal to 36.86989 and so on.

Now remember, Iโm not actually asked to find ๐ผ, Iโm asked to find ๐. So I need to subtract this value from 180 because theyโll sit on a straight line together. So Iโm gonna keep this value for ๐ผ 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 ๐ is equal to 180 minus this value of 36.869 and so on. So then ๐ is equal to 143.13. The question asked me to give ๐ to the nearest degree. So I need to round my answer. So my final answer then is that ๐ is equal to 143 degrees. And of course, that fits with ๐ being an obtuse angle.

So in summary then, in this video, weโve seen an application of the tangent ratio to calculating the angle that exists between a line segment and the ๐ฅ-axis.