if (0,-7) lies on the terminal side of angle theta in standard position, find each value sec(theta) , cot(theta) and sin(theta)

(0,0),(9,-7).

x = 0, Y = -7.
r^2 = X^2 + Y^2,
r^2 = 0 + (-7)^2 = 49,
r = 7.

secA = r/X = 7/0 = Infinite.

cotA = X/Y = 0 / -7 = 0.

sinA = Y/r = -7 / 7 = -1.

To find the values of sec(theta), cot(theta), and sin(theta) when the point (-7,0) lies on the terminal side of angle theta in standard position, we need to determine the values of the trigonometric functions from the given coordinates.

First, let's analyze the coordinates (-7,0). The x-coordinate is -7, which corresponds to the adjacent side of the angle theta, and the y-coordinate is 0, which corresponds to the opposite side of the angle theta.

To find the hypotenuse, we can use the Pythagorean theorem. Since the y-coordinate is 0, the opposite side has a length of 0. Therefore, we only need to calculate the length of the adjacent side.

Using the Pythagorean theorem, we have:
adjacent^2 + opposite^2 = hypotenuse^2

(-7)^2 + 0^2 = hypotenuse^2
49 = hypotenuse^2
hypotenuse = sqrt(49)
hypotenuse = 7

Now we have the lengths of the adjacent side (x-coordinate) and the hypotenuse. With these values, we can find the values of sec(theta), cot(theta), and sin(theta).

sec(theta) = hypotenuse / adjacent
sec(theta) = 7 / (-7)
sec(theta) = -1

cot(theta) = adjacent / opposite
cot(theta) = -7 / 0

However, cot(theta) is undefined when the opposite side is 0. Therefore, cot(theta) is undefined in this case.

sin(theta) = opposite / hypotenuse
sin(theta) = 0 / 7
sin(theta) = 0

Therefore, sec(theta) = -1, cot(theta) is undefined, and sin(theta) = 0.

Please note that when the opposite side is 0, cot(theta) is undefined, and when the adjacent side is 0, cosec(theta) is undefined. These are important conditions to consider when calculating trigonometric functions.