__Questions__

How can this mountaineer draw an accurate map?

How did sailors navigate before GPS?

How can computers rotate images so that they can be viewed from many different directions?

__Answer__

They look at angles and sides of triangles and make connections between them.

This is called **trigonometry.** A lot of the time, trigonometry is applied in 3D, and this is the skill that you will practise in this activity.

**Trigonometry ratios - recap**

You should be able to recall, for any right-angled triangle:

**sin x = O/H**

**cos x = A/H**

**tan x = O/A**

When you are dealing with a 3D shape, you will need to draw a right-angled triangle inside it and identify the relevant lengths to apply sin, cos, or tan.

__Example__

*Shown below is cuboid ABCDEFGH.*

**AB** = 20cm

**BD** = 4cm

**DH** = 5cm

*Calculate angle ABE.*

** Step 1:** Sketch the shape and label the lengths if they are not already written down.

We were given **AB** = 20cm, **BD** = 4cm, and **DH** = 5cm, so we should identify these sides and write the correct lengths.

__ Step 2:__ Draw a right-angled triangle onto the shape that contains the angle you want to find.

In this case, we want angle **ABE**, so our triangle must consist of points **A**, **B**, and **E**.

** Step 3:** Re-draw the triangle in 2D and use the correct trig ratio to solve.

The question seems much more simple if we just look at our triangle separately.

We want angle **ABE**, labelled **x**. The adjacent length = 20cm and the opposite length = 5cm.

We must therefore use **tan** to solve.

**tan x = O/A**

**tan x = 5/20 = 0.25**

**x = tan ^{-1} (0.25)**

**x = 14.0º (1 d.p.)**

Now that you have seen a worked example, have a go at calculating angles in other 3D shapes. Make sure you sketch the shape, identify the triangle you need, and use the correct trig ratio.