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 { 7 \pi } { 2 } \\) \\( \sin \theta = \\) \\( \csc \theta = \\) \\( \cos \theta = \\) \\( \sec \theta = \\) \\( \tan \theta = \\) \\( \cot \theta = \\)

Explanation:

Step1: Find \(\sin\theta\)

On the unit - circle, for \(\theta=\frac{7\pi}{2}\), which is equivalent to \(\theta = 3\pi+\frac{\pi}{2}\). The \(y\) - coordinate (which gives \(\sin\theta\)) for \(\theta=\frac{7\pi}{2}\) is \(- 1\), so \(\sin\theta=-1\).

Step2: Find \(\csc\theta\)

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

Step3: Find \(\cos\theta\)

The \(x\) - coordinate (which gives \(\cos\theta\)) for \(\theta=\frac{7\pi}{2}\) is \(0\), so \(\cos\theta = 0\).

Step4: Find \(\sec\theta\)

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

Step5: Find \(\tan\theta\)

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

Step6: Find \(\cot\theta\)

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

Answer:

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