(b) given \\( \\sin ( \\theta ) = - \\frac { \\sqrt { 3 } } { 2 } \\) and \\( \\cos ( \\theta ) > 0 \\)…

Explanation:

Step1: Determine the quadrant

Since (\sin(\theta)=-\frac{\sqrt{3}}{2}<0) and (\cos(\theta)>0), (\theta) is in the fourth quadrant.

Step2: Find (\cos(\theta))

Using the identity (\sin^{2}\theta+\cos^{2}\theta = 1), we have (\cos\theta=\sqrt{1-\sin^{2}\theta}). Substitute (\sin\theta=-\frac{\sqrt{3}}{2}) into the formula: [ \begin{align*} \cos\theta&=\sqrt{1-\left(-\frac{\sqrt{3}}{2}\right)^{2}}\ &=\sqrt{1-\frac{3}{4}}\ &=\sqrt{\frac{1}{4}}\ &=\frac{1}{2} \end{align*} ]

Step3: Calculate (\tan(\theta))

Using the formula (\tan\theta=\frac{\sin\theta}{\cos\theta}), substitute (\sin\theta =-\frac{\sqrt{3}}{2}) and (\cos\theta=\frac{1}{2}): [ \tan\theta=\frac{-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=-\sqrt{3} ]

Answer:

(-\sqrt{3})