complete the following table using exact values. do not rationalize any denominators.\nquestion help: video…

Explanation:

Step1: Recall the reciprocal relationship

We know that (\csc x=\frac{1}{\sin x}).

Step2: Calculate (\csc0^{\circ})

Since (\sin0^{\circ} = 0), (\csc0^{\circ}=\frac{1}{\sin0^{\circ}}), division by zero is undefined.

Step3: Calculate (\csc30^{\circ})

Given (\sin30^{\circ}=\frac{1}{2}), then (\csc30^{\circ}=\frac{1}{\sin30^{\circ}}=\frac{1}{\frac{1}{2}} = 2).

Step4: Calculate (\csc45^{\circ})

Given (\sin45^{\circ}=\frac{1}{\sqrt{2}}), then (\csc45^{\circ}=\frac{1}{\sin45^{\circ}}=\frac{1}{\frac{1}{\sqrt{2}}}=\sqrt{2}).

Step5: Calculate (\csc60^{\circ})

Given (\sin60^{\circ}=\frac{\sqrt{3}}{2}), then (\csc60^{\circ}=\frac{1}{\sin60^{\circ}}=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}).

Step6: Calculate (\csc90^{\circ})

Given (\sin90^{\circ} = 1), then (\csc90^{\circ}=\frac{1}{\sin90^{\circ}}=\frac{1}{1}=1).

Answer:

(x)	(\sin x)	(\csc x)
(0^{\circ})	(0)	undefined
(30^{\circ})	(\frac{1}{2})	(2)
(45^{\circ})	(\frac{1}{\sqrt{2}})	(\sqrt{2})
(60^{\circ})	(\frac{\sqrt{3}}{2})	(\frac{2}{\sqrt{3}})
(90^{\circ})	(1)	(1)