why does supplementary angles always have the same sin?
use the difference formula:
sin(π-x) = sinπ cosx - cosπ sinx
sinπ=0 and cosπ = -1, so you wind up with
0 - (-1)(sinx) = sinx
Also, if you draw the angles in standard position, recall that
sinθ = y/r
cosθ = x/r
since y is positive in QI and QII, the two angles have the same sine value.
This also shows why cos(π-x) = -cosx -- x is negative in QII.
Supplementary angles are defined as two angles whose sum is 180 degrees. The reason why supplementary angles always have the same sine is because the sine function is symmetric about the line of symmetry at 90 degrees.
To understand this concept, let's consider two supplementary angles: angle A and angle B, where A + B = 180 degrees. The sine function is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle.
Now, let's draw a unit circle, which is a circle with a radius of 1 unit. The x-axis represents the cosine values, and the y-axis represents the sine values of the angles in the unit circle.
For angle A, we can associate it with a point (x1, y1) on the unit circle. The x-coordinate (x1) would represent the cosine value, and the y-coordinate (y1) would represent the sine value of angle A.
When we consider angle B, which is the supplement of angle A, it would be located on the opposite side of the unit circle. Therefore, the x-coordinate (x2) of angle B would be equal to the negative of the x-coordinate (x1) of angle A, and the y-coordinate (y2) of angle B would be equal to the negative of the y-coordinate (y1) of angle A.
Since the sine function reflects the y-coordinate about the x-axis, the sine value of angle B (y2) would be equal to the negative of the sine value of angle A (y1). In other words, sin(B) = -sin(A).
Therefore, for any two supplementary angles A and B, their sine values will always have opposite signs, but the same magnitudes. This means that sin(A) = -sin(B) or sin(B) = -sin(A).
In conclusion, supplementary angles always have the same sine value because the sine function is symmetric about the line of symmetry at 90 degrees on the unit circle.