Final answer:
The six trigonometric functions for the given point (-1, -3) on the terminal side of an angle are sin = -3/√10, cos = -1/√10, tan = 3, csc = √10/-3, sec = √10/-1, and cot = 1/3. To find these, use the Pythagorean theorem to compute the hypotenuse and then calculate each function from the point's coordinates.
Step-by-step explanation:
To find all six trigonometric functions of an angle, given a point on its terminal side, we first identify the point, which in this case is (-1, -3). The x-coordinate (-1) represents the adjacent side, the y-coordinate (-3) represents the opposite side, and the hypotenuse can be found using the Pythagorean theorem. The hypotenuse (h) would be the square root of the sum of the squares of x and y, so h = √((-1)^2 + (-3)^2) = √(1 + 9) = √10.
The six trigonometric functions can then be found as follows:
- Sine (sin) is the opposite over hypotenuse, so sin = -3/√10.
- Cosine (cos) is the adjacent over hypotenuse, so cos = -1/√10.
- Tangent (tan) is the opposite over adjacent, so tan = -3/(-1) = 3.
- Cosecant (csc) is the reciprocal of sine, so csc = √10/-3.
- Secant (sec) is the reciprocal of cosine, so sec = √10/-1.
- Cotangent (cot) is the reciprocal of tangent, or the adjacent over opposite, so cot = -1/(-3) = 1/3.
If any function results in division by zero, then that trigonometric function is undefined.