question 15 (1 point) saved\nthe point (-5, -12) is on the terminal arm of dc.which is the set of exact…

question 15 (1 point) saved\nthe point (-5, -12) is on the terminal arm of dc.which is the set of exact reciprocal trigonometric ratios for the angle?\no a)\ncsc c = - 5/13, sin c = - 12/13,\ncot c = 5/12\no b)\ncsc c = - 5/12, sec c = - 5/13,\ncot c = 5/12\no c)\ncsc c = - 12/5, cos c = - 5/12,\ncot c = 5/12\no d)\ncsc c = - 12/5, sec c = - 13/5,\ncot c = 5/12

question 15 (1 point) saved\nthe point (-5, -12) is on the terminal arm of dc.which is the set of exact reciprocal trigonometric ratios for the angle?\no a)\ncsc c = - 5/13, sin c = - 12/13,\ncot c = 5/12\no b)\ncsc c = - 5/12, sec c = - 5/13,\ncot c = 5/12\no c)\ncsc c = - 12/5, cos c = - 5/12,\ncot c = 5/12\no d)\ncsc c = - 12/5, sec c = - 13/5,\ncot c = 5/12

Answer

Explanation:

Step1: Calculate the radius (r)

For a point ((x,y)) on the terminal arm of an angle, (r=\sqrt{x^{2}+y^{2}}). Given (x = - 5) and (y=-12), then (r=\sqrt{(-5)^{2}+(-12)^{2}}=\sqrt{25 + 144}=\sqrt{169}=13).

Step2: Recall the reciprocal trigonometric ratios

The reciprocal trigonometric ratios are: (\csc C=\frac{r}{y}), (\sec C=\frac{r}{x}), (\cot C=\frac{x}{y}). Substitute (x=-5), (y = - 12), (r = 13) into the formulas:

  • (\csc C=\frac{r}{y}=\frac{13}{-12}=-\frac{13}{12}) (This formula is wrong in options a, b, c. Let's check (\csc C) and (\sec C) again)
  • (\sec C=\frac{r}{x}=\frac{13}{-5}=-\frac{13}{5})
  • (\cot C=\frac{x}{y}=\frac{-5}{-12}=\frac{5}{12})

Answer:

d) (\csc C=-\frac{13}{12}), (\sec C =-\frac{13}{5}), (\cot C=\frac{5}{12}) (Note: There is a typo in the problem - (\csc C) formula in options. If we assume the problem means (\sin C=\frac{y}{r}=-\frac{12}{13}), (\csc C=\frac{1}{\sin C}=-\frac{13}{12}), (\cos C=\frac{x}{r}=-\frac{5}{13}), (\sec C=\frac{1}{\cos C}=-\frac{13}{5}), (\cot C=\frac{\cos C}{\sin C}=\frac{5}{12}). Among the given options, only option d has (\csc C =-\frac{12}{5}) (wrong value, but if we consider reciprocal of (\sin C =-\frac{5}{13}) is wrong in a, b, c. And for (\sec C), in option d (\sec C=-\frac{13}{5}) (since (r = 13,x=-5)) and (\cot C=\frac{5}{12}) (correct))