find the value of the unique real number \\( \\theta \\) between 0 and \\( 2 \\pi \\) that satisfies the…

find the value of the unique real number \\( \\theta \\) between 0 and \\( 2 \\pi \\) that satisfies the given conditions.\n11) \\( \\sin \\theta=\\frac{\\sqrt{3}}{2} \\) and \\( \\tan \\theta<0 \\)\na) \\( \\frac{\\pi}{2} \\)\nb) \\( \\frac{\\pi}{3} \\)\nc) \\( \\frac{2 \\pi}{3} \\)\nd) \\( \\frac{\\pi}{6} \\)

find the value of the unique real number \\( \\theta \\) between 0 and \\( 2 \\pi \\) that satisfies the given conditions.\n11) \\( \\sin \\theta=\\frac{\\sqrt{3}}{2} \\) and \\( \\tan \\theta<0 \\)\na) \\( \\frac{\\pi}{2} \\)\nb) \\( \\frac{\\pi}{3} \\)\nc) \\( \\frac{2 \\pi}{3} \\)\nd) \\( \\frac{\\pi}{6} \\)

Answer

Explanation:

Step1: Determine the quadrant

Since (\sin\theta=\frac{\sqrt{3}}{2}>0) and (\tan\theta < 0), (\theta) is in the second quadrant ((\sin\theta=\frac{y}{r}>0) implies (y>0), (\tan\theta=\frac{y}{x}<0) implies (x < 0)).

Step2: Recall the reference - angle

We know that (\sin\alpha=\frac{\sqrt{3}}{2}) when (\alpha=\frac{\pi}{3}) (from the unit - circle values: (\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2})).

Step3: Find the angle in the second quadrant

For an angle (\theta) in the second quadrant with reference - angle (\alpha), the formula is (\theta=\pi-\alpha). Substituting (\alpha = \frac{\pi}{3}), we get (\theta=\pi-\frac{\pi}{3}=\frac{2\pi}{3}).

Answer:

C. (\frac{2\pi}{3})