How to calculate sin without a calculator

In higher mathematics, we often notice that some things which are really easy to talk about but difficult to express rigorously have a property which is really easy to express rigorously but something that we probably wouldn't have thought of to begin with.

The trig functions are one of these things. With (a lot of) effort, you can show that

where the patterns of increasing the powers of x by 2 and switching between + and − signs continues forever. (The denominators also have a pattern: take the power that x is raised to in the term and multiply it by all of the smaller numbers down to 1; that is the number in the denominator). Note that you have to use radians and not degrees for this exact formula to work.

Of course, we can't sit around multiply and add for the rest of our lives just to compute sin 1, but we can just cut off the operations after a couple terms. If you go out to the x7 term, you can guarantee that your answer is accurate to at least 3 decimal places as long as you use angles between -π/2 and π/2.