### Trigonometry and 3D shapes

Working with 3D shapes can be made easy - by working with them in 2D.

Given a 3D shape, you can imagine cutting the shape up to give you a slice, which is flat along where you cut it (and 2D!). Working in 2D makes it much easier.

It is the same idea as when we look at Pythagoras in 3D, but now we look at general triangles, since you know how to calculate lengths and areas using trigonometry.

For example

Look at the following trapezium

[picture missing]

Calculate the angle between BD and BE.

First, consider the triangle BDE. Recognise that this is not a right-angled triangle. But we can use trigonometry.

We can easily work out length BE and BD using pythagoras.

BE is the hypotenuse from the triangle BCE: BE2 = 72 + 32 = 7.62

BD is the hypotenuse from the triangle BCD: BD2 = 72 + 32 = 7.62

DE is given: DE = 3

We know 3 sides, and we want to know an angle, so we can use the Cosine Rule: a2 = b2 + c2 – 2bc(Cos A)

We must label DE side a, because angle A is opposite DE.

a = DE = 3
b = 7.62
c = 7.62

Now we just have to plug numbers in!

9 = 58 + 58 -116(cos A)

0.922 = cos A

A = cos-1(0.992) = 22.71

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