Question

question 4 (1 point)
the point (3, -5) is on the terminal arm of ∠d.which is the set of exact primary
trigonometric ratios for the angle?

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

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

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

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

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 \(x = 3\) and \(y=-5\), then \(r=\sqrt{3^{2}+(-5)^{2}}=\sqrt{9 + 25}=\sqrt{34}\).

Step2: Calculate \(\sin D\)

The formula for \(\sin\theta=\frac{y}{r}\). Substituting \(y=-5\) and \(r = \sqrt{34}\), we get \(\sin D=\frac{-5}{\sqrt{34}}\).

Step3: Calculate \(\cos D\)

The formula for \(\cos\theta=\frac{x}{r}\). Substituting \(x = 3\) and \(r=\sqrt{34}\), we get \(\cos D=\frac{3}{\sqrt{34}}\).

Step4: Calculate \(\tan D\)

The formula for \(\tan\theta=\frac{y}{x}\). Substituting \(y=-5\) and \(x = 3\), we get \(\tan D=\frac{-5}{3}\).

Answer:

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