Final answer:
The exact values of cos θ, csc θ, and tan θ for the point (-7,3) are -7/√58, √58/3, and -3/7 respectively. We use the definitions of trigonometric ratios and the Pythagorean theorem to find these values.
Step-by-step explanation:
The point (-7,3) lies on the terminal side of an angle θ in standard position. To find the trigonometric ratios like cos θ, csc θ, and tan θ, we consider this point as representing the coordinates (x,y) on the Cartesian plane and interpret it in the context of a right triangle where x = -7, y = 3, and the hypotenuse r can be found using the Pythagorean theorem: r = √(x² + y²).
Therefore, the hypotenuse r is √((-7)² + (3)²) = √(49 + 9) = √58. Now, we use the definitions of the trigonometric functions in terms of a right triangle:
- cos θ = x/r = -7/√58
- csc θ = 1/sin θ = r/y = √58/3
- tan θ = y/x = 3/(-7) = -3/7
Thus, the exact values of the trigonometric functions for the given point are: cos θ = -7/√58, csc θ = √58/3, and tan θ = -3/7.