Final answer:
To find sinθ, cosθ, and tanθ for the angle with the point −6,5∙ on its terminal side, we use the x-coordinate as the adjacent side, the y-coordinate as the opposite side, and the distance from the origin to the point as the hypotenuse. The calculations yield sinθ = 5/√61, cosθ = -6/√61, and tanθ = -5/6.
Step-by-step explanation:
The point −6,5∙ is on the terminal side of an angle θ in standard position. The values of sine (θ), cosine (θ), and tangent (θ) are ratios that can be derived from a right triangle with this point as one vertex and the origin as another. In a right triangle, the length of the side adjacent to θ is the x-coordinate, the length of the side opposite to θ is the y-coordinate, and the hypotenuse is the distance from the origin to the point (-6, 5) which can be calculated using the Pythagorean theorem.
First, calculate the hypotenuse: hypotenuse = √((-6)^2 + 5^2) = √(36 + 25) = √61.
Thus, sinθ = opposite/hypotenuse = 5/√61 and cosθ = adjacent/hypotenuse = -6/√61. The tanθ is the ratio of the opposite side to the adjacent side, so tanθ = 5/(-6) = -5/6.