Final answer:
The sine of θ is -8 / √73, the cosecant of θ is -√73 / 8, and the cotangent of θ is 3/8 for the point (-3, -8) on the terminal side of θ.
Step-by-step explanation:
To find the exact values of sin θ, cscθ, and cotθ for a point (-3, -8) on the terminal side of θ, we will use trigonometric definitions from right triangle properties. Consider the point (-3, -8) as a representation of the opposite (y-coordinate) and adjacent (x-coordinate) sides of a right triangle in the Cartesian plane. The hypotenuse can be found using the Pythagorean theorem, which is the square root of the sum of the squares of the opposite and adjacent sides, hence the hypotenuse (h) is √((-3)² + (-8)²) = √(9 + 64) = √73.
The sine function (θ) is defined as the ratio of the opposite side to the hypotenuse, so sin θ = -8 / √73. The cosecant (θ) is the reciprocal of the sine, so cscθ = -√73 / 8. The cotangent (θ) is the ratio of the adjacent side to the opposite side, so cotθ = -3 / -8 = 3/8.