Question

If someone could explain the thought process behind 1F-5 from http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/part-b-implicit-differentiation-and-inverse-functions/problem-set-2/MIT18_01SC_pset1prb.pdf that would be really helpful. Where does the -1^k part from this formula sin(π/2+ kπ)=(−1)^k come from?
(part of the solution @ http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/part-b-implicit-differentiation-and-inverse-functions/problem-set-2/MIT18_01SC_pset1sol.pdf)

Answer 1

k even, i.e. k=2n for some integer n, then :\[\sin(\pi/2+k \pi)= \sin(\pi/2 + 2n \pi )= 1\]k odd, i.e. k=2n-1 for some integer n:\[\sin(\pi/2+k \pi)= \sin(\pi/2 + (2n-1) \pi )= -1\] (Remember this from trig? You can see this on a graph of y=sin(theta))
We find the y-coordinates of the points by substituting the x-coordinates into sin(x)+sin(y)=1/2, and using the values calculated above. It turns out only some of the x-values will give real answers for y, so only these x-coordinates are kept. We are then left with the x-coordinates and y-coordinates that work - the points we wanted.