But how do you know that the sine of -pi/2 is equal to negative 1
The sine function can be defined using the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. To find the sine of an angle, you draw a line from the origin to the point on the unit circle that intersects the angle. The y-coordinate of that point is equal to the sine of the angle.
For -π/2 radians (or -90 degrees), if you draw a line from the origin to the unit circle at that angle, the point of intersection will be at (-1, 0). Since the y-coordinate is 0, the sine of -π/2 radians is 0.
However, if you are referring to the sine of -π/2 in complex numbers, it is a different result. In that case, Euler's formula comes into play. Euler's formula states that eiθ = cos(θ) + i sin(θ), where i is the imaginary unit. If θ is -π/2, then e-iπ/2 = cos(-π/2) + i sin(-π/2).
The cosine of -π/2 is 0, and the sine of -π/2 is -1, so e-iπ/2 simplifies to 0 + (-i)(1) = -i. Therefore, the sine of -π/2 in complex numbers is -i.