Question

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

1. \\( \sin \theta=\frac{\sqrt{3}}{2} \\) and \\( \tan \theta<0 \\)

a) \\( \frac{\pi}{2} \\)
b) \\( \frac{\pi}{3} \\)
c) \\( \frac{2 \pi}{3} \\)
d) \\( \frac{\pi}{6} \\)

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