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Simple Trigonometric Calculations Using the Shadow Box

When the alilade intersects the edge of the shadow box, it forms a triangle. The alilade on your astrolabe is labeled for use with the right hand shadow box (look for the half of the alilade labeled 5, 10 and 15.) You will notice that if you lay the alilade horizontally against the upper edge of the right hand shadow box, the width of the box as indicated by the alilade is ten. Now try aligning the alilade with the corner of the box. Note that the corner intersects the alilade at about 14.1. What you have just calculated is the square root of 200 using the Pythagorean theorem. The sum of the square of the two sides equals the square of the hypotenuse (i.e. 10^2 + 10^2 = 200 = 14.1^2.) The angle indicated by the altitude scale of the astrolabe is nothing more than the angle of a right triangle--perfect for calculating trigonometric functions. As you will see, the astrolabe allows you to turn trigonometric calculations into simple multiplication and division.

Example: What is the sine, cosine and tangent of 30 degrees? Align the edge of the rule with 30 degrees on the altitude scale in the lower right corner of the back of the astrolabe. The alilade will form a triangle in the right hand shadow box. The triangle (and the box) are ten units wide. The height of the triangle is 6 while the hypotenuse is about 11.5. Using ratios of these numbers, we can calculate sine, cosine and tangent as indicated in the figure below.