Find the sin(Θ) of an angle in standard position if the terminal side passes through the point (4, -8).

recall sinØ = y/r

from the given: r^2 = 4^2 + (-8)^2 = 80
r = √80 = 4√5

since (4,-8) is in quad IV , and in IV the sine is negative,
sinØ = -8/4√5 = -2/√5
or -2√5/5

To find the sin(Θ) of an angle in standard position, where the terminal side passes through a given point (x, y), you can follow these steps:

Step 1: Determine the hypotenuse of the right triangle formed by the given point (x, y) as well as the origin (0, 0). The hypotenuse is the distance between these two points, which can be calculated using the distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2). In this case, the origin is (0, 0), so the distance formula simplifies to d = √(x^2 + y^2).

Step 2: Determine the length of the side adjacent to the angle. Since the angle is in standard position, the side adjacent to the angle is the x-coordinate of the given point.

Step 3: Use the trigonometric identity sin(Θ) = opposite/hypotenuse to find the sine of the angle. In this case, the opposite side is the y-coordinate of the given point.

Let's apply these steps to solve the problem:

Step 1: The distance from the origin to the point (4, -8) is calculated as follows:
d = √(x^2 + y^2) = √((4)^2 + (-8)^2) = √(16 + 64) = √80

Step 2: The length of the side adjacent to the angle is the x-coordinate, which is 4.

Step 3: Use the sine identity sin(Θ) = opposite/hypotenuse. In this case:
sin(Θ) = y-coordinate/hypotenuse = -8/√80

Therefore, the sin(Θ) of the angle in standard position, where the terminal side passes through the point (4, -8), is -8/√80 (or approximately -0.8944).