Task 9 Let's make sure we believe this addition formula. (a) Let a = π/6 and b = 7/3, and check to see whether the equation above holds. (You should be able to do this without a calculator.) [3 points] (b) Test this again with a = π/6 and b = π/6000 (or some other very small angle). What do you notice about the two terms on the right side of the equation for the addition formula? [3 points] The second idea that Euler needed in order to find the derivative of sine involves representing transcendental functions (like sine) as the sum of infinitely many powers of r. To Euler, this was also part of precalculus, and he included details in his precalculus book called the Introductio in Analysin Infinitorum (or Introduction to the Analysis of the Infinite for short), but these days we usually teach this idea (called "Taylor Series") in our calculus classes. Happily, we don't need to know much about these fascinating function representations to follow what Euler did next. All we need to know is that x -+20 is really close to sin x - especially for small values of z! Similarly, cos x is very close to the value of 1-+- 6

Fig: 1