Trig Gallery

Here are some intresting values of sine, cosine and tangent. Some of them you will probably never see in a high school or maybe even college class room. They are more intresting to look at than they are useful. To me, they show the complexity of a function that at first glance seems so elemtry; the shape of a circle is not only infinite, but also contains beauty.

Here are all the exact values of sine, cosine and tangent from 0 to 14, 15 to 29 and 30 to 45 degrees, increment of 1 degree. They can all be derived using my tutorial on finding the Sine of 1 Degree using doubling and addition formulas for sine and cosine. These three tables were created using Mathematica 4.1 from the FunctionExpand and Simplify command. Please note that Mathematica uses a much more efficient algorithm to generate these values as well as an amazing Simplify function so do not frustraited if your answers do not look like these. As long as they both numerical evaluate to the same thing then they are right.

Here are all the exact values of the sine, cosine and tangent of Pi/x with x from 1 to 10. Most of these are complex values that are just to show that Mathematica (4.0 and above) is capable of doing. As you can see, prime numbers seem to issue the most complex values. Of course, multiples of these primes become even more complex. Intresting to note is when x == 17.

This is an algebraic number, given exactly with square roots and no complex numbers. I found out why this happens now: the expansion will produce these kinds of numbers when the divisor of a*pi (a being an integer) is a Fermat Prime. So, this occurs at 3,5,17,257,65537...

Here is the sin(pi/257) | Notebook.

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