### Video Transcript

In this video, weโre going to review the trigonometric ratios of sine, cosine, and tangent. And then weโll see how we can use these to find unknown sides or angles in three dimensions.

The first important thing to note is that our trig ratios apply in right triangles. And if weโre working with an angle of ๐, then the labeling of the sides of our right triangle is very important. The longest side on a right triangle is called the hypotenuse. And in some diagrams where itโs harder to see, then we could always find it by the fact that itโs opposite the right angle. The labeling of the other two sides depends on the position of the angle. The side thatโs opposite the angle in question is the opposite side. The side thatโs next to or adjacent to the angle is called the adjacent side.

We can define the trigonometric ratios for an angle ๐ in the following way. Sine, which is abbreviated so that it looks like โsinโ but still pronounced sine, is sin of ๐ equals the opposite over the hypotenuse. cosine, abbreviated to cos, gives us cos of ๐ equals adjacent over hypotenuse. Finally, tangent, abbreviated to tan ๐, is calculated by the opposite over the adjacent sides.

Itโs common to see SOH CAH TOA written as a way to help us remember these ratios. So when it comes to applying the trig ratios in 3D, we still follow the same principles, but we need to find right triangles. If, for example, we had this cuboid, then this triangle would be a right triangle in two dimensions, so would this triangle. But would this triangle be? No, it wouldnโt be. This triangle is not in two dimensions. It covers three dimensions. Very often when weโre solving trigonometry problems in three dimensions, we find that we need to use two or more different 2D triangles to help us find a missing length or angle.

We may also find that we need to use the Pythagorean theorem too. We recall that this tells us that the square of the hypotenuse is equal to the sum of the squares on the other two sides. Weโll now look at some trigonometry problems in three dimensions. And we remember that weโre looking for right triangles in two dimensions.

Using the trigonometric ratios, find tan of ๐.

In this question, we have this rectangular prism or cuboid, and we can see that ๐ is the angle between the line ๐น๐ท and ๐น๐บ. In order to use the trigonometric ratios, we need to have a right triangle. We could create a right triangle with the triangle ๐น๐ท๐บ and the right angle here at vertex ๐บ. Note that this triangle, ๐น๐ท๐บ, would be a two-dimensional triangle as it sits on the face of our cuboid.

Itโs often very helpful to draw our triangles separately so that we can visualize the problem. Weโd have vertex ๐ท at the top and ๐น and ๐บ at the base of this triangle. ๐ท๐บ is given on the diagram as four centimeters, and ๐น๐บ is three centimeters. The angle ๐ is the angle here at ๐ท๐น๐บ. When weโre using the trigonometric ratios, we often use the phrase SOH CAH TOA to help us remember them. The TOA part helps us to remember tan of the angle, so weโd have tan of ๐ equals the opposite over the adjacent sides.

The longest side or hypotenuse isnโt needed for the tan ratio. The side thatโs opposite the angle ๐ is the length ๐ท๐บ. The side thatโs adjacent to our angle ๐ is the length ๐น๐บ. So we begin by saying that tan ๐ equals O over A. Thatโs opposite over adjacent. And we fill in the lengths that weโre given. The opposite side is four centimeters, and the adjacent side is three centimeters. Our answer then for tan ๐ is the fraction four-thirds.

Note that we werenโt actually asked to calculate the size of the angle ๐ but just to find tan ๐. If we did want to find the value of ๐, weโd need to use the inverse tan function on our calculator.

In the next question, weโll see our first example where we need to use two different two-dimensional triangles to help us find a missing angle.

A cuboid is shown in the figure. Calculate the angle between ๐ต๐ป and ๐ป๐น. Give your answer to two decimal places.

Weโre asked to consider the length ๐ต๐ป here. That would be the space diagonal of the cuboid. The diagonal here would cross through three dimensions. The other length here is ๐ป๐น. Weโre asked to calculate the angle between ๐ต๐ป and ๐ป๐น. Letโs call this angle ๐. So how might we go about calculating this angle? Well, we can see that we have a triangle ๐ต๐ป๐น. And we could also say that this will be a right triangle. Since we have a cuboid, we know that the length ๐ต๐น will meet ๐ป๐น at 90 degrees.

We might be familiar with two types of mathematics that we can apply in right triangles. We have the Pythagorean theorem, and we have trigonometry. As we have an unknown angle here, then we know that at some point, weโll need to apply trigonometry here. So letโs take a closer look at this triangle ๐ต๐ป๐น. Weโre given that ๐ต๐น is 3.5 centimeters and the angle that we need to find out is this one at ๐ต๐ป๐น, which weโve called ๐.

We donโt quite have enough information in order to be able to use trigonometry. Weโd need to know the length of the hypotenuse ๐ต๐ป or the length of the other side ๐ป๐น. So before we can attempt to find ๐, weโll need to find one of these other two lengths. Letโs have a look at this length ๐ป๐น. We can form another triangle ๐ป๐บ๐น on the base of this cuboid. We also know that this triangle too would be a right triangle.

So hereโs our other triangle drawn. We can see that ๐น๐บ on the diagram is three centimeters and ๐ป๐บ is given as four centimeters. Remember that itโs this side, ๐ป๐น, that we wish to find out. So on our pink diagram, we have this length here. So letโs define this with the letter ๐ฅ. When we have a right triangle, two sides that we know, and one side that we wish to find out, we can use the Pythagorean theorem. This tells us that the square on the hypotenuse is equal to the sum of the squares on the other two sides.

The first step then to finding our unknown ๐น๐ป and then to find the angle is to apply the Pythagorean theorem. The longest side here, our hypotenuse, is ๐ฅ. And the other two sides are three and four. And it doesnโt matter which way round we write these. So we have ๐ฅ squared equals three squared plus four squared. As three squared is nine and four squared is 16, weโll have that ๐ฅ squared is equal to 25. To find the value of ๐ฅ, we take the square root of both sides of our equation. So ๐ฅ is the square root of 25. And thatโs equal to five centimeters.

We have now found the value that ๐ป๐น is five centimeters. So we can then go ahead and find our angle ๐. In order to work out which of our trigonometric ratios we need to use out of sine, cosine, or tangent, we need to look carefully at the sides that we have and wish to find out. We have the side opposite our angle ๐. And we have the side thatโs adjacent to it. Notice that for our hypotenuse, the longest side, we donโt have the value of it, and we donโt want to calculate it.

Using the phrase SOH CAH TOA, we can see that we have the O for opposite and the A for adjacent. So that means that we need our tan ratio. We can write this trig ratio as tan ๐ equals opposite over adjacent. We can then plug in the values that we have for the opposite and adjacent and solve for the angle. This gives us tan of ๐ equals 3.5 over five. In order to find ๐ then, we need to use the inverse tan. As weโre asked to give our answer to two decimal places, we can reasonably use a calculator here.

This inverse function of tan on our calculator is usually found above the tan button. Pressing shift or second function will allow us to type in this calculation. We can obtain the value that ๐ equals 34.99202 and so on. Rounding our answer to two decimal places means that we check our third decimal digit to see if itโs five or more. And so our answer is that the angle between ๐ต๐ป and ๐ป๐น is 34.99 degrees to two decimal places.

Letโs look at another question.

The figure shows a right triangular prism. Find the angle between ๐ด๐น and ๐ด๐ถ, giving your answer to two decimal places.

The first thing we can do in this question is identify our two lengths ๐ด๐น and ๐ด๐ถ. The length ๐ด๐น will cut across the rectangular face here. The length ๐ด๐ถ will be the diagonal of the base of this right triangular prism. The angle between them will be the angle ๐น๐ด๐ถ created here. And we can define this as the angle ๐. We can create a triangle ๐ด๐น๐ถ in order to help us calculate our unknown angle ๐.

Letโs take a closer look at this triangle ๐ด๐น๐ถ. We know that the length ๐น๐ถ is four centimeters and our angle here is ๐ at ๐น๐ด๐ถ. In order to apply trigonometry in this triangle, we need to be sure if we have a right triangle. Weโre told in the question that this is a right triangular prism, which means that we have a right angle here at angle ๐ธ๐ต๐ด and at ๐น๐ถ๐ท. And since this plane on the bottom, ๐ต๐ด๐ท๐ถ, meets our other plane ๐ธ๐ต๐ถ๐น at right angles, then we know that we have a right angle here at ๐น๐ถ๐ด.

We note, however, that as we look at our triangle ๐ด๐น๐ถ, we donโt quite have enough information. Weโre going to need to find the length of one of these other two sides. If, for example, we look at ๐ด๐ถ, we can see on our diagram that this length ๐ด๐ท, which is eight centimeters, is different to the length of ๐ด๐ถ. Weโll need to create another right triangle in two dimensions to help us find the length of this side ๐ด๐ถ.

We can, in fact, create this triangle ๐ด๐ถ๐ท, which will have a right angle at angle ๐ด๐ท๐ถ. We can draw out triangle ๐ด๐ถ๐ท on the bottom of our right triangular prism. We can see that ๐ถ๐ท is three centimeters and ๐ด๐ท is eight centimeters. Donโt worry if your diagrams arenโt perfectly accurate; they donโt have to be to scale. Theyโre just there to help us visualize the problem. Remember why weโre doing this. Weโre trying to find the length ๐ด๐ถ, which is common to both triangles. We can define this as anything, but letโs call it the letter ๐ฅ.

When we find this length ๐ฅ on our first triangle, we can fill in the information into our second triangle. As we have a right triangle and two known sides and one unknown side, we can apply the Pythagorean theorem, which tells us that the square of the hypotenuse is equal to the sum of the squares on the other two sides. So we take our Pythagorean theorem, often written as ๐ squared equals ๐ squared plus ๐ squared. The hypotenuse, ๐, is our length ๐ฅ. So weโll have ๐ฅ squared equals three squared plus eight squared. Thatโs the length of our two other sides. And it doesnโt matter which way round we write those.

We can evaluate three squared is nine, eight squared is 64, and adding those gives us ๐ฅ squared equals 73. To find ๐ฅ, we take the square root of both sides of our equation. So we have ๐ฅ equals the square root of 73. Itโs very tempting at this point to pick up our calculator and find a decimal answer for the square root of 73. But as we havenโt finished with this value, weโre going to keep it in this square root form.

Now that we have found ๐ฅ, that means weโve found our length of ๐ด๐ถ. And so we can use this to find our angle ๐. As weโre interested in the angle here, that means weโre not going to use the Pythagorean theorem again, but weโll need to use some trigonometry. In order to work out which of the sine, cosine, or tangent ratios we need, weโll need to look at the sides that we have.

The length ๐น๐ถ is opposite our angle ๐. ๐ด๐ถ is adjacent to the angle. And the hypotenuse is always the longest side. Now, weโre not given the hypotenuse, and weโre not interested in calculating it, so we can remove it from this problem. Using SOH CAH TOA can be useful to help us figure out which ratio we want. We have the opposite and the adjacent sides, so that means that weโre going to use the tan or tangent ratio. tan of ๐ is given by the opposite over the adjacent sides. We now fill in the values that we have.

The opposite length is four centimeters, and the adjacent length is given by root 73. So tan of ๐ is equal to four over root 73. In order to find ๐ by itself, we need the inverse operation to tan. And thatโs finding the inverse tan. This function, written as tan with a superscript negative one, can usually be found on our calculator above the tan button. Using our calculator to evaluate this will give us ๐ equals 25.0873 and so on. And the units here will be degrees as, of course, this is an angle, not a length.

Weโre asked to round our answer to two decimal places. So that means we check our third decimal digit to see if itโs five or more. And as it is, then our answer is given as 25.09 degrees. And so the angle between ๐ด๐น and ๐ด๐ถ is 25.09 degrees.

Before we finish with this question, letโs just review what we could have done instead. When we started this question, we had this large triangle ๐ด๐น๐ถ which cut through our triangular prism. We were told that ๐น๐ถ was four centimeters. And we worked out this length of ๐ด๐ถ. But could we have done it by working out the length of ๐ด๐น instead?

If we look at our triangular prism, in order to work out the length of ๐ด๐น, weโd need another triangle. We do have a right triangle here. And this length of ๐ธ๐น will be eight centimeters. However, if we were trying to work out this length of ๐ด๐น, weโd also need to work out this length of ๐ด๐ธ. In order to find ๐ด๐ธ, weโd need to create yet another right triangle. We would know that ๐ด๐ต is four centimeters and ๐ด๐ต is also three centimeters. So we would eventually get the correct answer for ๐ด๐น and, therefore, our angle ๐. Itโs just that that second method would involve three triangles instead of the two triangles that we used.

Letโs now summarize what weโve learned in this video. Firstly, we reviewed our trigonometric ratios. sin of ๐ is equal to the opposite over hypotenuse sides. cos of ๐ equals the adjacent over the hypotenuse. And the tan of ๐ equals the opposite over adjacent sides. All of these are valid for an angle ๐ in a right triangle. These ratios can be easily recalled by using the phrase SOH CAH TOA.

When using the trigonometric ratios in three dimensions, we need to be using two-dimensional right triangles to help us. And we often need to use more than one of these two-dimensional triangles. A handy tip is that if we are using more than one triangle, we should keep our answers to our calculation in the square root form until we get to our final answer where we can round the decimal value if required.

And finally, as we saw in a few questions, when weโre solving trigonometry in three dimensions, we might also require the Pythagorean theorem as well. In the same way as for our trigonometric ratios in 3D, the same applies for the Pythagorean theorem in 3D. That is, we need to be using it in two-dimensional right triangles only.