In a Unit circle, which has a radius of one unit, Coordinates of every point on the circumference are determined in terms of Cosine and Sine of the angle which is formed by the radius at that point with the x-axis. For example if P is any point on the circumference of the unit circle, with centre at O, such that line OP makes an angle θ with the x axis, then coordinates of P will be ( Cos θ , Sin θ ) as shown in the figure below.

Negative angles are measured in Clockwise direction starting with Positive x axis. Angle -9π/2 would be represented by the Line in Red Pen on the negative Y axis. The coordinates of the corresponding point on the circumference are clearly (0, -1). Since the corresponding angle is -9 π/2, Cos -9 π/2 would be represented by 0 and Sin-9 π/2 would be represented by -1. Hence using unit circle, we have determined the value of Sin -9 π/2.

Again using Trig Identities, Sin -9 π/2 will be - Sin9 π/2 . Write 9 π/2 as 4π + π/2. This would gives us -Sin9 π/2 = - Sin π/2 = -1 [Sin π/2 equals 1 ]

For using a triangle to determine Sin -9 π/2 or, -Sin π/2, consider a Right Triangle with a fixed length perpendicular and let the base contract, so that hypotenuse moves closure to the Perpendicular line. If the base ultimately gets reduced to zero, the Perpendicular and Hypotenuse would superimpose on each other and the base angle will become a right angle.

Thus Sin π/2 would be perpendicular/ hypotenuse = 1.

Hence Sin-9 π/2 = - Sin9 π/2= -Sin π/2= -1

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