### Video Transcript

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.