Suppose that the point P(13,-30) is on the terminal side of the angle theta in standard position. What is sin(theta)? Give your answer to the nearest thousandth

P is in the 4th quadrant, so sin = -30/sqrt(1069) = -0.918

To find sin(theta), we need to determine the length of the side opposite to the angle theta and the length of the hypotenuse of the right triangle formed in the standard position.

Given that the point P(13, -30) is on the terminal side of the angle theta, we can form a right triangle where the vertical leg is -30 (representing the y-coordinate of point P) and the horizontal leg is 13 (representing the x-coordinate of point P).

To find the length of the hypotenuse, we can use the Pythagorean theorem:

hypotenuse^2 = vertical leg^2 + horizontal leg^2

Substituting the given values:

hypotenuse^2 = (-30)^2 + 13^2
hypotenuse^2 = 900 + 169
hypotenuse^2 = 1069

Taking the square root of both sides, we find the length of the hypotenuse:

hypotenuse ≈ √1069
hypotenuse ≈ 32.699

Now that we have the lengths of the vertical leg (-30) and the hypotenuse (32.699), we can find sin(theta) by dividing the vertical leg by the hypotenuse:

sin(theta) = vertical leg / hypotenuse
sin(theta) = -30 / 32.699
sin(theta) ≈ -0.918

Therefore, sin(theta) is approximately -0.918.