How to prove sin (-x) = -sin(x) for any angle of ‘x’ , w/o applying quadrant norms (ALL,SIN,TAN,COS)? Please help

Question

How to prove sin (-x) = -sin(x) for any angle of ‘x’ , w/o applying quadrant norms (ALL,SIN,TAN,COS)? Please  help

anonymous · Answer

sin(a) is the y value in the unit circle of angle a (see diagram)
So if we take sin(-a), we will always be on the other side of the y axis. (-a means 360-a°)
Therefore is sin(a) is positive, sin(-a) will be the negative value of sin(a) and the opposite applies

anonymous · Answer

Also we can use the sum/diff formula for sin
sin(-a) = sin(0-a) = sin(0)cos(a)-cos(0)sin(a)
                           = 0*cos(a)-1*sin(a)
                           = -sin(a)

anonymous · Answer

apple_pi thanks