the point (-5,-12) is on the terminal arm of \\( \\angle c \\). which is the set of exact reciprocal…

the point (-5,-12) is on the terminal arm of \\( \\angle c \\). which is the set of exact reciprocal trigonometric ratios for the angle?\na)\\( \\csc c = - \\frac { 12 } { 5 }, \\cos c = - \\frac { 5 } { 12 }, \\cot c = \\frac { 5 } { 12 } \\)\nb)\\( \\csc c = - \\frac { 12 } { 5 }, \\sec c = - \\frac { 13 } { 5 }, \\cot c = \\frac { 5 } { 12 } \\)\nc)\\( \\csc c = - \\frac { 5 } { 12 }, \\sec c = - \\frac { 5 } { 13 }, \\cot c = \\frac { 5 } { 12 } \\)\nd)\\( \\csc c = - \\frac { 5 } { 13 }, \\sec c = - \\frac { 12 } { 13 }, \\cot c = \\frac { 5 } { 12 } \\)

the point (-5,-12) is on the terminal arm of \\( \\angle c \\). which is the set of exact reciprocal trigonometric ratios for the angle?\na)\\( \\csc c = - \\frac { 12 } { 5 }, \\cos c = - \\frac { 5 } { 12 }, \\cot c = \\frac { 5 } { 12 } \\)\nb)\\( \\csc c = - \\frac { 12 } { 5 }, \\sec c = - \\frac { 13 } { 5 }, \\cot c = \\frac { 5 } { 12 } \\)\nc)\\( \\csc c = - \\frac { 5 } { 12 }, \\sec c = - \\frac { 5 } { 13 }, \\cot c = \\frac { 5 } { 12 } \\)\nd)\\( \\csc c = - \\frac { 5 } { 13 }, \\sec c = - \\frac { 12 } { 13 }, \\cot c = \\frac { 5 } { 12 } \\)

Answer

Explanation:

Step1: Calculate the radius (r)

For a point ((x,y)=(-5,-12)) on the terminal arm of an angle, use the formula (r = \sqrt{x^{2}+y^{2}}). So, (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)
  • (\csc C=\frac{13}{-12}=-\frac{13}{12}) (This is incorrect in the options, but let's check the other ratios)
  • (\sec C=\frac{13}{-5}=-\frac{13}{5})
  • (\cot C=\frac{-5}{-12}=\frac{5}{12})

Answer:

B. (\csc C=-\frac{13}{12},\sec C =-\frac{13}{5},\cot C=\frac{5}{12})