the point (3, -5) is on the terminal arm of ∠d. which is the set of exact primary trigonometric ratios for…

the point (3, -5) is on the terminal arm of ∠d. which is the set of exact primary trigonometric ratios for the angle?\n a) sin d = -\\frac{\\sqrt{34}}{3}, cos d = \\frac{\\sqrt{34}}{5}, tan d = -\\frac{5}{3}\n b) sin d = \\frac{\\sqrt{34}}{5}, cos d = -\\frac{\\sqrt{34}}{3}, tan d = -\\frac{5}{3}\n c) sin d = -\\frac{\\sqrt{34}}{5}, cos d = \\frac{\\sqrt{34}}{3}, tan d = -\\frac{5}{3}\n d) sin d = -\\frac{\\sqrt{34}}{5}, cos d = \\frac{\\sqrt{34}}{3}, tan d = -\\frac{1}{3}\n

the point (3, -5) is on the terminal arm of ∠d. which is the set of exact primary trigonometric ratios for the angle?\n a) sin d = -\\frac{\\sqrt{34}}{3}, cos d = \\frac{\\sqrt{34}}{5}, tan d = -\\frac{5}{3}\n b) sin d = \\frac{\\sqrt{34}}{5}, cos d = -\\frac{\\sqrt{34}}{3}, tan d = -\\frac{5}{3}\n c) sin d = -\\frac{\\sqrt{34}}{5}, cos d = \\frac{\\sqrt{34}}{3}, tan d = -\\frac{5}{3}\n d) sin d = -\\frac{\\sqrt{34}}{5}, cos d = \\frac{\\sqrt{34}}{3}, tan d = -\\frac{1}{3}\n

Answer

Explanation:

Step1: Calculate the radius ( r )

For a point ((x,y)) on the terminal side of an angle, (r=\sqrt{x^{2}+y^{2}}). Given (x = 3) and (y=-5), then (r=\sqrt{3^{2}+(-5)^{2}}=\sqrt{9 + 25}=\sqrt{34}).

Step2: Calculate (\sin D), (\cos D) and (\tan D)

The definitions of trigonometric functions are (\sin D=\frac{y}{r}), (\cos D=\frac{x}{r}) and (\tan D=\frac{y}{x}). Since (x = 3), (y=-5) and (r=\sqrt{34}), we have:

  • (\sin D=\frac{y}{r}=\frac{-5}{\sqrt{34}}=-\frac{5\sqrt{34}}{34}) (rationalize the denominator: (\frac{-5}{\sqrt{34}}\times\frac{\sqrt{34}}{\sqrt{34}}=-\frac{5\sqrt{34}}{34}))
  • (\cos D=\frac{x}{r}=\frac{3}{\sqrt{34}}=\frac{3\sqrt{34}}{34}) (rationalize the denominator: (\frac{3}{\sqrt{34}}\times\frac{\sqrt{34}}{\sqrt{34}}=\frac{3\sqrt{34}}{34}))
  • (\tan D=\frac{y}{x}=\frac{-5}{3})

Answer:

b) (\sin D =-\frac{5\sqrt{34}}{34},\cos D=\frac{3\sqrt{34}}{34},\tan D =-\frac{5}{3})