How do you use the unit circle to derive #sin(2pi-theta)#?

To derive sin(2π - θ) using the unit circle, we start by considering the trigonometric identity sin(π - θ) = sin(θ). Since the unit circle represents the values of sine and cosine for angles from 0 to 2π, we can observe that the sine function is symmetric about the y-axis. Therefore, sin(π - θ) is equal to sin(θ) but with the opposite sign.

Now, we apply this concept to sin(2π - θ). Since 2π - θ is the same as π - (θ - π), we can rewrite sin(2π - θ) as sin(π - (θ - π)). Applying the identity sin(π - θ) = sin(θ), we get sin(2π - θ) = sin(θ - π).

Now, we determine the coordinates of the point on the unit circle corresponding to the angle (θ - π). The x-coordinate of this point is equal to cos(θ - π) and the y-coordinate is equal to sin(θ - π).

Since the unit circle is symmetric about the y-axis, the cosine of an angle θ - π is the same as the negative of the cosine of the angle -θ + π. Therefore, cos(θ - π) = -cos(-θ + π).

Similarly, the sine of an angle θ - π is the same as the negative of the sine of the angle -θ + π. Therefore, sin(θ - π) = -sin(-θ + π).

Now, substituting these values back into sin(2π - θ) = sin(θ - π), we get sin(2π - θ) = -sin(-θ + π). Therefore, sin(2π - θ) = -sin(-θ + π).

In conclusion, using the unit circle and trigonometric identities, we can derive sin(2π - θ) as -sin(-θ + π).