The point P lies on the terminal side of the angle. Find the value of the trigonometric function: 𝑃(−(5),5) ,𝑐𝑜𝑡(Θ)

Answer 1

Here are two approaches to the problem:

1. Consider that (-5,5) on a circle of radius 5 would have the same angle as (-1,1) on the unit circle. Both points lie on the diagonal line y = -x, which is 45° to the left of the y-axis in the second quadrant. Add 90° for the angle between the y-axis and x-axis in the first quadrant because the angle we use in trig functions starts from the positive x-axis. So Θ = 90° + 45° = 135° (or 3π / 4 radians). Then note that cot(Θ) = 1 / tan(Θ). You can calculate that tan(3π /4) = -1. So cot(3π /4) = 1 / tan(3π /4) = 1 / (-1) = -1.
2. You could also consider the right triangle formed by the points (-5,5), (-5,0), and (0,0). Even without finding the angle, we know that cot(Θ) = x/y = -5/5 = -1. To help with your understanding, think about why cotangent is always negative in the second quadrant.

Remember that cot(Θ) = cos(Θ) / sin(Θ) = x/y where (x,y) can be thought of as a point on a circle. The angle Θ starts from the positive x-axis and goes counterclockwise around the circle to the terminal point (x,y).