Final answer:
The six trigonometric functions for the point (-5, -1) can be found by considering this point as part of a right triangle on the Cartesian plane. The hypotenuse is calculated using the Pythagorean theorem, which allows us to find the sin, cos, tan, csc, sec, and cot of the angle associated with that point.
Step-by-step explanation:
The six trigonometric functions for a point (-5, -1) on the Cartesian plane can be determined using the definitions of sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot) in relation to a right triangle. To do this, we consider the given point as part of a right triangle with the origin (0,0), where the x-coordinate (-5) represents the length of the side adjacent to the angle, and the y-coordinate (-1) represents the length of the opposite side. The hypotenuse can be calculated using the Pythagorean theorem and is the distance from the origin (0,0) to the point (-5, -1), which is √((-5)² + (-1)²) = √(26).
Thus, the six trigonometric functions are calculated as follows:
- sin θ = opposite/hypotenuse = -1/√(26)
- cos θ = adjacent/hypotenuse = -5/√(26)
- tan θ = opposite/adjacent = -1/-5 = 1/5
- csc θ (cosecant) = hypotenuse/opposite = √(26)/-1
- sec θ (secant) = hypotenuse/adjacent = √(26)/-5
- cot θ (cotangent) = adjacent/opposite = -5/-1 = 5