Let (7,-3) be a point on the terminal side of theta. Find the exact values of cos of theta, sec of theta and cot of theta?

To find the exact values of the trigonometric functions for theta, we can use the given point (7, -3) on the terminal side.

First, we need to determine the values of the adjacent side, opposite side, and hypotenuse based on the coordinates of the point.

The x-coordinate (7) corresponds to the adjacent side, and the y-coordinate (-3) corresponds to the opposite side.

Using the Pythagorean theorem, we can find the value of the hypotenuse (r):

r = √(adjacent^2 + opposite^2)
= √(7^2 + (-3)^2)
= √(49 + 9)
= √58

Now that we have the values of the adjacent side (7), opposite side (-3), and hypotenuse (√58), we can find the trigonometric ratios for theta.

Cosine (cos) of theta is defined as the ratio of the adjacent side to the hypotenuse:

cos(theta) = adjacent / hypotenuse
= 7 / √58

Secant (sec) of theta is the reciprocal of cosine:

sec(theta) = 1 / cos(theta)
= 1 / (7 / √58)
= √58 / 7

Cotangent (cot) of theta is defined as the ratio of the adjacent side to the opposite side:

cot(theta) = adjacent / opposite
= 7 / (-3)
= -7/3

Therefore, the exact values of cos(theta), sec(theta), and cot(theta) are:

cos(theta) = 7 / √58
sec(theta) = √58 / 7
cot(theta) = -7/3

To find the exact values of cos(theta), sec(theta), and cot(theta) given that (7, -3) is a point on the terminal side of theta, we need to determine the values of the trigonometric functions using the given point.

First, we can assign values to the sides of a right triangle using the coordinates of the given point. The horizontal side would correspond to the x-coordinate, which is 7, and the vertical side would correspond to the y-coordinate, which is -3. The hypotenuse can be found using the Pythagorean theorem.

Using the Pythagorean theorem:
Hypotenuse^2 = (Horizontal side)^2 + (Vertical side)^2

Hypotenuse^2 = 7^2 + (-3)^2
Hypotenuse^2 = 49 + 9
Hypotenuse^2 = 58

Taking the square root of both sides:
Hypotenuse = √58

Now, we can find the values of the trigonometric functions:

1. Cosine (cos(theta)) is equal to the ratio of the adjacent side (x-coordinate) to the hypotenuse:
cos(theta) = adjacent / hypotenuse
cos(theta) = 7 / √58
To simplify this, we can rationalize the denominator by multiplying both the numerator and the denominator by √58:
cos(theta) = 7√58 / (√58 * √58)
cos(theta) = 7√58 / 58
Therefore, the exact value of cos(theta) is 7√58 / 58.

2. Secant (sec(theta)) is the reciprocal of cosine, so it is the ratio of the hypotenuse to the adjacent side:
sec(theta) = hypotenuse / adjacent
sec(theta) = √58 / 7

Therefore, the exact value of sec(theta) is √58 / 7.

3. Cotangent (cot(theta)) is the ratio of the adjacent side (x-coordinate) to the opposite side (y-coordinate):
cot(theta) = adjacent / opposite
cot(theta) = 7 / -3
Therefore, the exact value of cot(theta) is -7/3.

To summarize:
exact value of cos(theta) = 7√58 / 58
exact value of sec(theta) = √58 / 7
exact value of cot(theta) = -7/3

r^2= 7^2 + (-3)^2 = 58

r = √58

since (7,-3) is in IV,
cosØ = 3/√58
secØ = √58/3
cotØ = -7/3