sin a = sin 180-a-please prove or explain?

Question

sin a = sin 180-a please prove or explain
(a=alpha)

Answer 1

sin (180 - α) = sin 180 cos α - cos 180 sinα
sin(180 - α) = 0 cos α + 1 sin α
sin (180 - α) = sin α

Answer 2

Draw a unit circle centered at the origin of an x-y coordinate system. Pick a point on the circle. The coordinates of the point will be (cos(a),sin(a)). This means your y coordinate is the sine of angle a.

Play around with angles and you'll see that doing the subtraction negates the cos(a) part but leaves the sin(a) part with the same sign.

Answer 3

The sine of any angle between 0 degrees & 180 degrees is positive because y is positive. Between 180 degrees and 0 degrees, it is negative because y is negative. The sines of any two angles 180 degrees apart are the same, except one is negative, because y, or the distance from where the radius intercepts the circumference to the x-axis is the same.

Answer 4

sin (A - B) = sin A cosB - cos A sin B
substitute A = 180 , B = a
sin(180 - a) = sin(180) cos a - cos(180) sin(a)
= (0)cos(a) - (-1)sin(a) [since sin 180 = 0 and cos(180) = -1]
= 0 + sin (a)
sin(180 - a) = sin (a)