question 7 - 5 marks. what is the exact value of \\( \\csc 300^{\\circ} \\)? a) \\( \\frac{\\sqrt{3}}{2} \\)…

Explanation:

Step1: Use the co - function identity

We know that (\csc\theta=\frac{1}{\sin\theta}). So, (\csc300^{\circ}=\frac{1}{\sin300^{\circ}}).

Step2: Find the reference angle

The angle (300^{\circ}) is in the fourth quadrant. The reference angle (\alpha = 360^{\circ}-300^{\circ}=60^{\circ}). And in the fourth quadrant, (\sin\theta<0). So, (\sin300^{\circ}=-\sin60^{\circ}).

Step3: Calculate (\sin60^{\circ})

Since (\sin60^{\circ}=\frac{\sqrt{3}}{2}), then (\sin300^{\circ}=-\frac{\sqrt{3}}{2}).

Step4: Calculate (\csc300^{\circ})

(\csc300^{\circ}=\frac{1}{\sin300^{\circ}}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}). Rationalizing the denominator: (-\frac{2}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3})

Answer:

C. (-\frac{2\sqrt{3}}{3})