What are the exact values of each trigonometric function for the point P(9, 12) located on the terminal side of an angle of 8 radians?

To find the exact values of the trigonometric functions for the angle whose terminal side passes through the point P(9, 12), we will use the definitions of sine, cosine, and tangent based on the coordinates of the point.

First, we need to calculate the radial distance (r) from the origin to the point P(9, 12). The formula for this is:

r = √(x² + y²)

Substituting the coordinates of point P:

r = √(9² + 12²) = √(81 + 144) = √225 = 15

Now, we can find each of the trigonometric functions:

1. Sine

The sine function is defined as:

sin(θ) = opposite / hypotenuse

For our point P(9, 12):

sin(θ) = 12 / 15 = 4/5

2. Cosine

The cosine function is defined as:

cos(θ) = adjacent / hypotenuse

For our point P:

cos(θ) = 9 / 15 = 3/5

3. Tangent

Tangent is the ratio of sine to cosine:

tan(θ) = sin(θ) / cos(θ)

Thus:

tan(θ) = (12/15) / (9/15) = 12 / 9 = 4/3

4. Cosecant

The cosecant is the reciprocal of sine:

csc(θ) = 1 / sin(θ)

Hence:

csc(θ) = 5/4

5. Secant

The secant is the reciprocal of cosine:

sec(θ) = 1 / cos(θ)

This gives us:

sec(θ) = 5/3

6. Cotangent

The cotangent is the ratio of cosine to sine:

cot(θ) = cos(θ) / sin(θ)