Final answer:
To find the trigonometric values of angle t with the terminal side passing through (-3,-8), calculate the hypotenuse using the Pythagorean theorem, then determine each trigonometric function using the corresponding ratio involving x, y, and the hypotenuse.
Step-by-step explanation:
If the point (-3,-8) is on the terminal side of angle t, we can find all six trigonometric values for t by considering this point to represent the coordinates (x,y) in the Cartesian plane, and placing it with respect to a unit circle centered at the origin. In this context, the x and y values represent the lengths of the adjacent and opposite sides of a right triangle, respectively, and the hypotenuse can be found using the Pythagorean theorem.
First, we find the length of the hypotenuse r which is the distance from the origin to the point (-3,-8):
r = √((-3)² + (-8)²) = √(9 + 64) = √73
Now we can find the six trigonometric functions as follows:
- Sine (sin(t)) = opposite/hypotenuse = -8/√73
- Cosine (cos(t)) = adjacent/hypotenuse = -3/√73
- Tangent (tan(t)) = opposite/adjacent = -8/-3 = 8/3
- Cosecant (csc(t)) = hypotenuse/opposite = √73/-8
- Secant (sec(t)) = hypotenuse/adjacent = √73/-3
- Cotangent (cot(t)) = adjacent/opposite = -3/-8 = 3/8
It is important to note that the signs of these ratios are consistent with the quadrant in which the terminal side lies, which in this case is quadrant III, where both x and y values are negative.