in which quadrant would \\( \\theta \\) be if \\( \\tan \\theta = \\frac { \\sqrt { 3 } } { 3 } \\) and \\(…

in which quadrant would \\( \\theta \\) be if \\( \\tan \\theta = \\frac { \\sqrt { 3 } } { 3 } \\) and \\( \\sin \\theta = - \\frac { 1 } { 2 } \\)? use a digit for your response rather than a roman numeral. (1 point)\nquadrant\ncheck answer remaining attempts : 3

in which quadrant would \\( \\theta \\) be if \\( \\tan \\theta = \\frac { \\sqrt { 3 } } { 3 } \\) and \\( \\sin \\theta = - \\frac { 1 } { 2 } \\)? use a digit for your response rather than a roman numeral. (1 point)\nquadrant\ncheck answer remaining attempts : 3

Answer

Explanation:

Step1: Recall the signs of trigonometric functions in each quadrant

  • In Quadrant I: (\sin\theta> 0), (\cos\theta>0), (\tan\theta>0)
  • In Quadrant II: (\sin\theta> 0), (\cos\theta<0), (\tan\theta<0)
  • In Quadrant III: (\sin\theta< 0), (\cos\theta<0), (\tan\theta>0)
  • In Quadrant IV: (\sin\theta< 0), (\cos\theta>0), (\tan\theta<0)

Step2: Analyze the given trigonometric values

Given (\tan\theta=\frac{\sqrt{3}}{3}>0) and (\sin\theta =-\frac{1}{2}<0) Since (\tan\theta>0), (\theta) is in either Quadrant I or Quadrant III. Since (\sin\theta<0), (\theta) is in either Quadrant III or Quadrant IV.

Step3: Find the common quadrant

The common quadrant for (\tan\theta>0) and (\sin\theta<0) is Quadrant III.

Answer:

(3)