Question

standard 14 homework
score: 14.5/30 answered: 11/18
question 11
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if \\( \theta=\frac{-11 \pi}{6} \\), evaluate each of the following trigonometric expressions. give exact answers in fraction
form; no decimals allowed.
\\( \sec (\theta)= \\)
\\( \csc (\theta)= \\)
\\( \tan (\theta)= \\)
\\( \cot (\theta)= \\)
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Explanation:

Step1: Find the coterminal angle

Since \(\theta=\frac{-11\pi}{6}\), we can find a coterminal angle by adding \(2\pi\). \(\frac{-11\pi}{6}+2\pi=\frac{-11\pi + 12\pi}{6}=\frac{\pi}{6}\)

Step2: Evaluate \(\csc(\theta)\)

We know that \(\csc\theta=\frac{1}{\sin\theta}\), and \(\sin(\frac{\pi}{6})=\frac{1}{2}\), so \(\csc(\theta)=\frac{1}{\sin(\frac{\pi}{6})} = 2\)

Step3: Evaluate \(\tan(\theta)\)

We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), \(\sin(\frac{\pi}{6})=\frac{1}{2}\), \(\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}\), so \(\tan(\theta)=\frac{\sin(\frac{\pi}{6})}{\cos(\frac{\pi}{6})}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}\)

Step4: Evaluate \(\cot(\theta)\)

We know that \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), \(\sin(\frac{\pi}{6})=\frac{1}{2}\), \(\cos(\frac{\pi}{6})=\frac{\sqrt{3}}{2}\), so \(\cot(\theta)=\frac{\cos(\frac{\pi}{6})}{\sin(\frac{\pi}{6})}=\sqrt{3}\)

Answer:

\(\csc(\theta)=2\), \(\tan(\theta)=\frac{1}{\sqrt{3}}\), \(\cot(\theta)=\sqrt{3}\)