Question

complete the following table using exact values. do not rationalize any denominators.

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Explanation:

Step1: Recall the reciprocal identity

The secant function is the reciprocal of the cosine function, i.e., \(\sec x=\frac{1}{\cos x}\)

Step2: Calculate \(\sec x\) for \(x = 0^{\circ}\)

When \(x = 0^{\circ}\), \(\cos x=1\). Then \(\sec x=\frac{1}{\cos x}=\frac{1}{1} = 1\)

Step3: Calculate \(\sec x\) for \(x = 30^{\circ}\)

When \(x = 30^{\circ}\), \(\cos x=\frac{\sqrt{3}}{2}\). Then \(\sec x=\frac{1}{\cos x}=\frac{2}{\sqrt{3}}\)

Step4: Calculate \(\sec x\) for \(x = 45^{\circ}\)

When \(x = 45^{\circ}\), \(\cos x=\frac{1}{\sqrt{2}}\). Then \(\sec x=\frac{1}{\cos x}=\sqrt{2}\)

Step5: Calculate \(\sec x\) for \(x = 60^{\circ}\)

When \(x = 60^{\circ}\), \(\cos x=\frac{1}{2}\). Then \(\sec x=\frac{1}{\cos x}=2\)

Step6: Calculate \(\sec x\) for \(x = 90^{\circ}\)

When \(x = 90^{\circ}\), \(\cos x = 0\). Then \(\sec x=\frac{1}{\cos x}\), and division by zero is undefined. So \(\sec90^{\circ}\) is undefined

Answer:

\(x\)	\(\cos x\)	\(\sec x\)
\(30^{\circ}\)	\(\frac{\sqrt{3}}{2}\)	\(\frac{2}{\sqrt{3}}\)
\(45^{\circ}\)	\(\frac{1}{\sqrt{2}}\)	\(\sqrt{2}\)
\(60^{\circ}\)	\(\frac{1}{2}\)	\(2\)
\(90^{\circ}\)	\(0\)	Undefined