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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$$\begin{array}{|c|c|}\\hline \\sin \\theta = & \\csc \\theta = \\\\ \\hline \\cos \\theta = & \\sec \\theta = \\\\ \\hline \\tan \\theta = & \\cot \\theta = \\\\ \\hline \\end{array}$$

$

Explanation:

Step1: Recall unit - circle values

On the unit circle, for \(\theta=\frac{\pi}{2}\), the coordinates are \((x,y)=(0,1)\).
The definitions of trigonometric functions: \(\sin\theta = y\), \(\cos\theta=x\), \(\tan\theta=\frac{y}{x}\), \(\csc\theta=\frac{1}{y}\), \(\sec\theta=\frac{1}{x}\), \(\cot\theta=\frac{x}{y}\)

Step2: Calculate \(\sin\theta\)

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

Step3: Calculate \(\csc\theta\)

Since \(\csc\theta=\frac{1}{y}\) and \(y = 1\), \(\csc\theta = 1\)

Step4: Calculate \(\cos\theta\)

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

Step5: Calculate \(\sec\theta\)

Since \(\sec\theta=\frac{1}{x}\) and \(x = 0\), \(\sec\theta=\text{DNE}\)

Step6: Calculate \(\tan\theta\)

Since \(\tan\theta=\frac{y}{x}\) and \(x = 0\), \(\tan\theta=\text{DNE}\)

Step7: Calculate \(\cot\theta\)

Since \(\cot\theta=\frac{x}{y}\) and \(x = 0,y = 1\), \(\cot\theta=0\)

Answer:

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