Question Video: Finding an Angle Using Trigonometry in 3D Mathematics

A cuboid is shown in the figure. Calculate the angle between ๐ต๐ป and ๐ป๐น. Give your answer to two decimal places.

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Video Transcript

A cuboid is shown in the figure. Calculate the angle between ๐ต๐ป and ๐ป๐น. Give your answer to two decimal places.

Weโ€™re asked to consider the length ๐ต๐ป here. That would be the space diagonal of the cuboid. The diagonal here would cross through three dimensions. The other length here is ๐ป๐น. Weโ€™re asked to calculate the angle between ๐ต๐ป and ๐ป๐น. Letโ€™s call this angle ๐œƒ. So how might we go about calculating this angle? Well, we can see that we have a triangle ๐ต๐ป๐น. And we could also say that this will be a right triangle. Since we have a cuboid, we know that the length ๐ต๐น will meet ๐ป๐น 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, weโ€™ll need to apply trigonometry here. So letโ€™s take a closer look at this triangle ๐ต๐ป๐น. Weโ€™re given that ๐ต๐น is 3.5 centimeters and the angle that we need to find out is this one at ๐ต๐ป๐น, which weโ€™ve called ๐œƒ.

We donโ€™t quite have enough information in order to be able to use trigonometry. Weโ€™d need to know the length of the hypotenuse ๐ต๐ป or the length of the other side ๐ป๐น. So before we can attempt to find ๐œƒ, weโ€™ll need to find one of these other two lengths. Letโ€™s have a look at this length ๐ป๐น. We can form another triangle ๐ป๐บ๐น on the base of this cuboid. We also know that this triangle too would be a right triangle.

So hereโ€™s our other triangle drawn. We can see that ๐น๐บ on the diagram is three centimeters and ๐ป๐บ is given as four centimeters. Remember that itโ€™s this side, ๐ป๐น, that we wish to find out. So on our pink diagram, we have this length here. So letโ€™s define this with the letter ๐‘ฅ. 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 ๐น๐ป and then to find the angle is to apply the Pythagorean theorem. The longest side here, our hypotenuse, is ๐‘ฅ. And the other two sides are three and four. And it doesnโ€™t matter which way round we write these. So we have ๐‘ฅ squared equals three squared plus four squared. As three squared is nine and four squared is 16, weโ€™ll have that ๐‘ฅ squared is equal to 25. To find the value of ๐‘ฅ, we take the square root of both sides of our equation. So ๐‘ฅ is the square root of 25. And thatโ€™s equal to five centimeters.

We have now found the value that ๐ป๐น is five centimeters. So we can then go ahead and find our angle ๐œƒ. 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œƒ. And we have the side thatโ€™s adjacent to it. Notice that for our hypotenuse, the longest side, we donโ€™t have the value of it, and we donโ€™t 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œƒ 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œƒ equals 3.5 over five. In order to find ๐œƒ then, we need to use the inverse tan. As weโ€™re 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œƒ equals 34.99202 and so on. Rounding our answer to two decimal places means that we check our third decimal digit to see if itโ€™s five or more. And so our answer is that the angle between ๐ต๐ป and ๐ป๐น is 34.99 degrees to two decimal places.

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