### Video Transcript

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.