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The Challenge: Create a Scale Map



How do we work out the distance to an object?

Trigonometric diagram of our calculationsTrigonometry is based around right angled triangles. So, to work out the distance to the object, we need to first make two right angled triangles.

First, draw a straight line on the ground 200m (about 219 yards) long. Stand at each end of the marked length and measure angles A and B.

You will now have two triangles with the distance to the object being a common side to both triangles (this is D).

Now one way to find out the distance to the object at right angles to the 200m line, distance D, is to find C1 and C2.

We use the formula the tangent of the angle (TAN)
= LENGTH OF THE OPPOSITE SIDE / LENGTH OF THE ADJACENT SIDE.

TAN A = D / C1 therefore D = C1 x TAN A

To work out C1, we can replace C2 in the equation with (200-C1).

These two equations can be re-written as:
D = C1 x TAN A and D = (200 - C1) x TAN B which is the equivalent to D = (200 x TAN B) - (C1 x TAN B)

The two equations can be combined to read:
C1 (TAN A + TAN B) = 200 x TAN B

To now find C1 we divide both halves of the equation by (TAN A + TAN B):
C1= 200 x TAN B / (TAN A + TAN B)

And from that: D= C1 x TAN A