Final answer:
To find the six trigonometric functions given a point (1, -8), calculate the hypotenuse using the Pythagorean theorem, then use the definitions of sine, cosine, tangent, cosecant, secant, and cotangent with the sides of the right triangle formed.
Step-by-step explanation:
To find the exact value of the six trigonometric functions of θ given a point (1, -8), you first need to understand the point represents a position on the Cartesian plane. The x-coordinate (1) is the value of the adjacent side to θ in a right triangle, and the y-coordinate (-8) is the value of the opposite side. To define the functions, we need the hypotenuse which we can find using the Pythagorean theorem: √(1² + (-8)²) = √(1 + 64) = √65.
The six trigonometric functions of θ based on the triangle with sides 1 (adjacent), -8 (opposite), and √65 (hypotenuse) are:
- Sine (θ) = Opposite / Hypotenuse = -8 / √65
- Cosine (θ) = Adjacent / Hypotenuse = 1 / √65
- Tangent (θ) = Opposite / Adjacent = -8 / 1 = -8
- Cosecant (θ) = Hypotenuse / Opposite = √65 / -8
- Secant (θ) = Hypotenuse / Adjacent = √65 / 1
- Cotangent (θ) = Adjacent / Opposite = 1 / -8 = -1/8
All functions are based on this right triangle and are determined using their respective trigonometric definitions with respect to the angle θ which lies in the fourth quadrant (since x is positive and y is negative).