in which quadrant would \\( \\theta \\) be if \\( \\tan \\theta=-\\sqrt{3} \\) and \\( \\sin \\theta>0 \\)…

Explanation:

Step1: Recall the sign of trigonometric functions in each quadrant

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

Step2: Analyze the given conditions

We are given that (\tan\theta =-\sqrt{3}<0) and (\sin\theta>0). From the sign - rules of trigonometric functions: If (\tan\theta=\frac{\sin\theta}{\cos\theta}<0), then (\sin\theta) and (\cos\theta) have opposite signs. Since (\sin\theta>0), then (\cos\theta<0)

Answer:

Quadrant II