If (-2,5) is a point on the terminal side of an angle theta, find the exact value of the six trigonometric functions of theta. Then find the actual measure of theta.

To find the exact values of the six trigonometric functions of theta, we need to determine the values of the opposite side, adjacent side, and hypotenuse.

Given that the point (-2, 5) is on the terminal side of theta, we can use this point to construct a right triangle. The x-coordinate (-2) will represent the adjacent side, while the y-coordinate (5) will represent the opposite side.

To find the hypotenuse, we can use the Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So, we have:

Adjacent side^2 + Opposite side^2 = Hypotenuse^2
(-2)^2 + (5)^2 = Hypotenuse^2
4 + 25 = Hypotenuse^2
29 = Hypotenuse^2

Now, we can take the square root of both sides to find the value of the hypotenuse:

Hypotenuse = √29

Now that we have determined the values for the opposite side, adjacent side, and hypotenuse, let's find the six trigonometric functions of theta:

1. Sine (sin): opposite/hypotenuse
sin(theta) = 5/√29

2. Cosine (cos): adjacent/hypotenuse
cos(theta) = -2/√29

3. Tangent (tan): opposite/adjacent
tan(theta) = 5/-2 = -5/2

4. Cosecant (csc): hypotenuse/opposite
csc(theta) = √29/5

5. Secant (sec): hypotenuse/adjacent
sec(theta) = √29/-2 = -√29/2

6. Cotangent (cot): adjacent/opposite
cot(theta) = -2/5

To find the actual measure of theta, we can use the information about the point (-2, 5) on the terminal side.

Since the point lies in the second quadrant, theta will have a reference angle in the first quadrant. Let's call this reference angle alpha.

Using the inverse sine function, we can determine the measure of the reference angle alpha:

alpha = sin^(-1)(5/√29)

Using symmetry, we know that the angle theta in the second quadrant will have the same value as 180 degrees minus the reference angle alpha:

theta = 180 - alpha

So, the actual measure of theta is theta = 180 - sin^(-1)(5/√29).