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

Using the trigonometric ratios, find tan ๐.

02:03

### Video Transcript

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 ๐น๐ท and ๐น๐บ. In order to use the trigonometric ratios, we need to have a right triangle. We could create a right triangle with the triangle ๐น๐ท๐บ and the right angle here at vertex ๐บ. Note that this triangle, ๐น๐ท๐บ, would be a two-dimensional triangle as it sits on the face of our cuboid.

Itโs often very helpful to draw our triangles separately so that we can visualize the problem. Weโd have vertex ๐ท at the top and ๐น and ๐บ at the base of this triangle. ๐ท๐บ is given on the diagram as four centimeters, and ๐น๐บ is three centimeters. The angle ๐ is the angle here at ๐ท๐น๐บ. When weโre 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 weโd have tan of ๐ equals the opposite over the adjacent sides.

The longest side or hypotenuse isnโt needed for the tan ratio. The side thatโs opposite the angle ๐ is the length ๐ท๐บ. The side thatโs adjacent to our angle ๐ is the length ๐น๐บ. So we begin by saying that tan ๐ equals O over A. Thatโs opposite over adjacent. And we fill in the lengths that weโre 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 werenโt 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, weโd need to use the inverse tan function on our calculator.