angle theta is in standard position on a co-ordinate grid. the terminal arm of theta is in quadrent 2 on the line that's equation is 3x-4y=0. Determine the sine of this angle and then the measure of angle theta.

4y = 3 x

y = .75 x + 0
slope = +.75
line is not in quadrant 2 but in one or three

To find the sine of the angle theta and the measure of angle theta, we first need to determine the coordinates of the point where the terminal arm intersects the line 3x - 4y = 0.

The equation 3x - 4y = 0 can be rewritten as y = (3/4)x, which represents a line with a slope of 3/4.

Since the terminal arm is in quadrant 2, both the x and y coordinates will be negative.

Let's assume the coordinates of the intersection point are (-x, -y).

Substituting -x and -y into the equation, we get:
3(-x) - 4(-y) = 0
-3x + 4y = 0

Now, we have a system of equations:
x = -x
3x - 4y = 0

From the first equation, we can determine that x = 0.

Substituting x = 0 into the second equation, we get:
0 - 4y = 0
-4y = 0
y = 0

Therefore, the coordinates of the intersection point are (0, 0).

Now, let's calculate the sine of the angle theta:
sin(theta) = y-coordinate / distance from the origin

Since the y-coordinate is 0 and the distance from the origin is the length of the hypotenuse, which is the distance from the point (0, 0) to the point where the terminal arm intersects the line.

Using the distance formula, the distance from the origin to (0, 0) is:
√((0 - 0)^2 + (0 - 0)^2) = √(0 + 0) = √0 = 0

Therefore, the sine of theta is 0.

To find the measure of angle theta, we can use the inverse sine function (arcsin).
theta = arcsin(0)
theta = 0

Hence, the sine of angle theta is 0, and the measure of angle theta is 0 degrees.