simplify this problem

Question

Simplify sin (180°-x)

Answer 1

Hey Mario! We could visualize a couple of "test" points:

x=0, sin(0)=0, sin(180-0)=sin(180)=0

x=90, sin(90)=1, sin(180-90)= sin(90)=1

x=-30,sin(-30)= -1/2, sin(180 minus -30)=sin(210)= -1/2 ... Looks like sin(180-x)= sin x ... Why so?

Visualize a clockface with 3:00 being the "zero" line. If /x is 2:00 for sin x, then /-x is 4:00 for -sin x ... What does the "180" offset do to 2:00? Makes it 8:00, or ANOTHER -sin x ... Graphically, the -x direction makes -sin x, and the 180 offset makes it negative AGAIN: minus(-sin x)= leaving plain 'ole sin x ... Regards :) 

Answer 2

To simplify this, we should use the difference formula for the sine function: sin(a-b)=sin(a)cos(b)-sin(b)cos(a) In our case it becomes: sin(180)cos(x)-sin(x)cos(180). Because sin(180)=0 and cos(180)=1 we have: 0*cos(x)-(-1)*sin(x)=sin(x). Thus sin(180-x)= sin(x).