Question

evaluate the six trigonometric function of $\theta$. (hint: sketch the angle and use your unit circle) if the ratio is undefined, enter dne.
$\theta=\frac{\pi}{2}$
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$$\begin{array}{|l|l|}\\hline\\sin\\theta= & \\csc\\theta= \\\\ \\hline\\cos\\theta= & \\sec\\theta= \\\\ \\hline\\tan\\theta= & \\cot\\theta= \\\\ \\hline\\end{array}$$

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Explanation:

Step1: Recall the unit - circle definitions

On the unit circle \(x = \cos\theta\), \(y=\sin\theta\), \(\tan\theta=\frac{y}{x}\), \(\csc\theta=\frac{1}{y}\), \(\sec\theta=\frac{1}{x}\), \(\cot\theta=\frac{x}{y}\). For \(\theta=\frac{\pi}{2}\), the point on the unit circle is \((x,y)=(0,1)\).

Step2: Calculate \(\sin\theta\)

Since \(y = \sin\theta\), when \(\theta=\frac{\pi}{2}\), \(\sin\theta = 1\).

Step3: Calculate \(\csc\theta\)

Since \(\csc\theta=\frac{1}{\sin\theta}\), and \(\sin\theta = 1\), then \(\csc\theta=\frac{1}{1}=1\).

Step4: Calculate \(\cos\theta\)

Since \(x=\cos\theta\), when \(\theta = \frac{\pi}{2}\), \(\cos\theta=0\).

Step5: Calculate \(\sec\theta\)

Since \(\sec\theta=\frac{1}{\cos\theta}\), and \(\cos\theta = 0\), division by zero is undefined, so \(\sec\theta=\text{DNE}\).

Step6: Calculate \(\tan\theta\)

Since \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and \(\cos\theta = 0\), division by zero is undefined, so \(\tan\theta=\text{DNE}\).

Step7: Calculate \(\cot\theta\)

Since \(\cot\theta=\frac{\cos\theta}{\sin\theta}\), and \(\cos\theta = 0\), \(\sin\theta=1\), then \(\cot\theta=\frac{0}{1}=0\).

Answer:

\(\sin\theta = 1\), \(\csc\theta = 1\), \(\cos\theta=0\), \(\sec\theta=\text{DNE}\), \(\tan\theta=\text{DNE}\), \(\cot\theta = 0\)