What is the relationship between sin(θ) and sin(180° − θ)?

We are asked to determine the relationship between sin(theta) and sin(180-theta) .
sin(theta)=sin(180-theta)
(1) Consider the unit circle. If theta is in the first quadrant, then 180-theta is in the second quadrant. Since in the unit circle sin(theta)=y we see that both theta and 180-theta will be positive. See the link below: the reference angle for the triangle associated with 180-theta is theta so the sin values are equal.
If theta is in the second quadrant then 180-theta is in the first quadrant and we get an analogous situation.
If theta is in the third quadrant (180If theta=0,90,180,270 it is easy to see that sin(theta)=sin(180-theta)
(2) We can use the difference formula: sin(A-B)=sinAcosB-sinBcosA
sin(180-theta)=sin180costheta-sinthetacos180
=0*costheta-sintheta*(-1)=sintheta
(3) On a graph of the sin function, we move 180 left and then reflect over the axis—this results in the same curve. (See attachment)
http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html