Method 2 - Using Trigonometry
In the previous activity you used a scale drawing and a protractor to measure the angles. An alternative method would be to use a dynamic geometry tool to make these measurements. (See the Sketchpad measurement lesson.)

You can also make these measurements by not measuring the angles at all; but, rather use something called Trigonometry to help you get these angles from the shadow lengths.

Looking at the triangles again, how can I determine the sun's angle without measuring it? Well, some clever person came up with a scheme that related angle measurements with the ratio of sides in a right triangle. For example,

the ratio of the gnomon (meter stick) and its shadow is 100/80.5 or 1.24. This ratio (gnomon divided by the shadow) has a name. It is called the tangent of the base angle.


Question: Is it possible to have a different (non-similar) right triangle where the gnomon is also 1.24 times larger than its shadow? 


Since a given ratio determines the angles, there are handy charts that make finding these angles convenient. Below is a table of tangents of angles. These are the corresponding ratios for the given angles. You will find that 1.24 lies between angles 51 and 52 degrees. (See Sketchpad version of tangent table.)

If the tangent of angle A is 1.24 then the base angle is a little more than 51 degrees and the sun angle is the complement of that angle which is between 38 and 39 degrees. 

In pre-technology days, you had to rely on Trig tables to find these values, but today you can also find this angle from a calculator or a spreadsheet.

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