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

Using the trigonometric ratios, find tan 𝜃.

02:03

### Video Transcript

Using the trigonometric ratios, find tan of 𝜃.

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

It’s often very helpful to draw our triangles separately so that we can visualize the problem. We’d have vertex 𝐷 at the top and 𝐹 and 𝐺 at the base of this triangle. 𝐷𝐺 is given on the diagram as four centimeters, and 𝐹𝐺 is three centimeters. The angle 𝜃 is the angle here at 𝐷𝐹𝐺. When we’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 we’d have tan of 𝜃 equals the opposite over the adjacent sides.

The longest side or hypotenuse isn’t needed for the tan ratio. The side that’s opposite the angle 𝜃 is the length 𝐷𝐺. The side that’s adjacent to our angle 𝜃 is the length 𝐹𝐺. So we begin by saying that tan 𝜃 equals O over A. That’s opposite over adjacent. And we fill in the lengths that we’re given. The opposite side is four centimeters, and the adjacent side is three centimeters. Our answer then for tan 𝜃 is the fraction four-thirds.

Note that we weren’t actually asked to calculate the size of the angle 𝜃 but just to find tan 𝜃. If we did want to find the value of 𝜃, we’d need to use the inverse tan function on our calculator.

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