Final answer:
To find the exact trigonometric function values for an angle, one must refer to the coordinates of the point through which the terminal side passes, form a right triangle, use the Pythagorean theorem to calculate the hypotenuse and then the six trigonometric ratios.
Step-by-step explanation:
To find the exact values of all trigonometric functions for an angle in standard position where the terminal side passes through a given point, let's use a right triangle that can be formed by dropping a perpendicular from the point to the x-axis. The x-coordinate represents the length of the adjacent side (Ax), the y-coordinate represents the length of the opposite side (Ay), and the distance from the origin to the point is the hypotenuse (A) of the right triangle. Using the Pythagorean theorem, we can determine the length of A from the coordinates Ax and Ay.
After computing A, we can find the trigonometric functions as follows:
Note that when Ax or Ay is zero, or if they are negative, the trigonometric functions will be undefined or will have both positive and negative values depending on the quadrant in which the angle lies.