To find the exact values of the trigonometric functions for the angle θ whose terminal side passes through the point P(9, 12), we will first determine the radius r from the origin to the point P.
The radius r can be calculated using the Pythagorean theorem:
r = √(x² + y²)
Where x = 9 and y = 12:
r = √(9² + 12²)
r = √(81 + 144)
r = √225
r = 15
Now that we have the radius, we can find the values of the trigonometric functions:
- sin(θ) = y/r
sin(θ) = 12/15 = 4/5 - cos(θ) = x/r
cos(θ) = 9/15 = 3/5 - tan(θ) = y/x
tan(θ) = 12/9 = 4/3 - csc(θ) = r/y
csc(θ) = 15/12 = 5/4 - sec(θ) = r/x
sec(θ) = 15/9 = 5/3 - cot(θ) = x/y
cot(θ) = 9/12 = 3/4
Thus, the exact values of the trigonometric functions are:
- sin(θ) = 4/5
- cos(θ) = 3/5
- tan(θ) = 4/3
- csc(θ) = 5/4
- sec(θ) = 5/3
- cot(θ) = 3/4