Triangles in the Blind-spot Box

Our new blind-spot measurer allows us to read off how far the eye is from the test card when the spot is in the blind spot, but what is the angle? It would be inconvenient to have to get a protractor out each time and measure the angle directly!

Fortunately, there is a simple relationship between the lengths of the sides of triangles ad the corresponding angles! This has been studied since ancient times (pyramids, timings of sacrifices etc…) and is now known as trigonometry. 

If you look at right-angled triangles with the small angle on the left...

If you put together 4 small triangles you make one big triangle that has all the same angles, and the sides are exactly twice as long. This means that for the big triangle the ratios of the sides are the same as for the little triangles. We can add more little triangles to make bigger and bigger triangles and the ratio of the upright side length to the bottom side length will stay the same. In fact it is the same for any triangle with these angles. So if we know the angles of a right-angled triangle, we know what the ratio of its sides will be (so long as we have measured that ratio on any similar triangle). Likewise, if we know the ratio of the edges, that will determine the angle.

This is just what we want! We know that the separation of the dot and cross is 7.4cm, and we measure the distance from the eye to the cross (it is marked on the box and add 1cm for the distance of the eye behind the window) so we can calculate the ratio. 

All we need is for a list of the ratio of these sides for every possible angle!

Fortunately (again) someone has done this for us! Islamic mathematicians invented the field of trigonometry in about the 9th century, and made tables of the ratios of triangle sides for all possible angles. These are SINE COSINE and TANGENT functions and you learn it in GCSE maths because it comes in useful in lots of areas of science. (Islamic mathematicians invented lots of the basic concepts of mathematics and lots of the words used are derived from Arabic).

Here’s the table we need:

Now all we need is 300 volunteers to measure!