-1 | …….|………|

-2 \——/\——/

where the x axis goes from 0 to 2 pi.

If k + | 2 sin x | = 0, then k must equal – |2 sin x|, the graph in #1.

A straight line that touches k only twice must pass through (pi/2, -2) and (3 pi / 2, -2) [ie, the bottom of the two curves]

y = -2

#3. shouldn’t there be a ‘y has 4 solutions’ ??? Y can range from y>-2 to y < 0. If y = -2, there are 2 solutions. If y = 0, there are 3 solutions. everywhere in between are 4 solutions. If you have Microsoft Excel, you can create a table of values, and get it to graph it for you: try a1 = 0 a2 = a1 +2 * PI () / 10 a3 = a2 + 2 * pi() / 10 etc... to a 11 b1 = - abs ( 2 * sin ( a1 ) ) b2 = -abs ( 2 * sin ( a2 )) etc. to b11 then use the chart wizard to do an x-y plot from a1 to b11