Answer Happy

1 e 1 5. The sine and cosine functions for complex input are defined by the for- mulas sin(2) (ez-e-), cos(z) = (ez te-i-). a. (2 pts) Assuming the trig identity cos(1 – 7/2) = sin(2) holds for all real r, use the definition of complex exponential e"+iv = e(cos(u) +i sin()) to show that the same identity cos(2 – */2) = sin(2) holds for all complex 2. b. (3 pts) Using the usual rules for exponents (expand both sides, show all algebra steps), prove the double-angle identity sin(22) = 2 cos(z) sin(2). c. (4 pts) Prove that for any w e C, there exists a number 2 € C so that sin(2) = u. Find an expression for 2 in terms of w and the possibly multi-valued) operations and log. d. (4 pts) Find infinitely many solutions z of the equation sin(2) = 5. Express your answer in terms of In, not sin-7. (Hints: use part c. to get or check an exact answer, although your calculator might help to find one approximate solution. You do not have to find every solution z, just infinitely many different ones.)