if (-12,5) is a pointon the terminal side of an angel 0 find the exact value of six trig funcation of 0

V(0,0), T(-12,5)

X = -12 - 0 = -12.
Y = 5- 0 = 5.
r = Sqrt(X^2+Y^2) = Sqrt(144+25) = 13

Sin A = Y/r = 5/13.
Cos A = X/r =
Tan A = Y/X

CSC A = r/Y
Sec A =
Cot A =

NOTES:
V = Vertex.
T = Terminal side.
Sqrt = Square root.

To find the exact values of the six trigonometric functions for angle θ, given that the point (-12, 5) lies on its terminal side, we need to use the coordinates of the point to determine the values of the trigonometric functions.

Let's start by calculating the hypotenuse of the right triangle formed by the given point (-12, 5), which lies on the terminal side of angle θ. The hypotenuse can be found using the Pythagorean theorem:

Hypotenuse = √((-12)^2 + 5^2)
= √(144 + 25)
= √(169)
= 13

Now we can determine the values of the trigonometric functions as follows:

1. Sine (sin θ) = opposite / hypotenuse
= 5 / 13

2. Cosine (cos θ) = adjacent / hypotenuse

To determine the adjacent side, we need to determine whether the angle is in quadrant II or III. Since the x-coordinate is negative (-12), the angle is in quadrant II. Therefore, the adjacent side is positive (√((-12)^2) = 12):

Cosine (cos θ) = 12 / 13

3. Tangent (tan θ) = opposite / adjacent
= 5 / 12

4. Cosecant (csc θ) = hypotenuse / opposite
= 13 / 5

5. Secant (sec θ) = hypotenuse / adjacent
= 13 / 12

6. Cotangent (cot θ) = adjacent / opposite
= 12 / 5

Therefore, the exact values of the six trigonometric functions of angle θ are as follows:

sin θ = 5/13
cos θ = 12/13
tan θ = 5/12
csc θ = 13/5
sec θ = 13/12
cot θ = 12/5