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

Using the trigonometric ratios, find tan ๐œƒ.


Video Transcript

Using the trigonometric ratios, find tan of ๐œƒ.

In this question, we have this rectangular prism or cuboid, and we can see that ๐œƒ 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œƒ 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œƒ 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œƒ is the length ๐ท๐บ. The side thatโ€™s adjacent to our angle ๐œƒ is the length ๐น๐บ. So we begin by saying that tan ๐œƒ 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œƒ is the fraction four-thirds.

Note that we werenโ€™t actually asked to calculate the size of the angle ๐œƒ but just to find tan ๐œƒ. If we did want to find the value of ๐œƒ, weโ€™d need to use the inverse tan function on our calculator.

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