Question

evaluate the six trigonometric function of $\theta$. (hint: sketch the angle and use your unit circle) if the ratio is undefined, enter dne.
$\theta = 0^{\circ}$
$\

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

$

Explanation:

Step1: Recall the definitions of trigonometric functions

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

Step2: Calculate \(\sin\theta\)

Since \(y = \sin\theta\) and \(y = 0\) when \(\theta=0^{\circ}\), \(\sin(0^{\circ})=0\).

Step3: Calculate \(\csc\theta\)

Since \(\csc\theta=\frac{1}{\sin\theta}\) and \(\sin(0^{\circ}) = 0\), \(\csc(0^{\circ})=\text{DNE}\) (division by zero is undefined).

Step4: Calculate \(\cos\theta\)

Since \(x=\cos\theta\) and \(x = 1\) when \(\theta = 0^{\circ}\), \(\cos(0^{\circ})=1\).

Step5: Calculate \(\sec\theta\)

Since \(\sec\theta=\frac{1}{\cos\theta}\) and \(\cos(0^{\circ})=1\), \(\sec(0^{\circ})=\frac{1}{1}=1\).

Step6: Calculate \(\tan\theta\)

Since \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) and \(\sin(0^{\circ}) = 0\), \(\cos(0^{\circ})=1\), \(\tan(0^{\circ})=\frac{0}{1}=0\).

Step7: Calculate \(\cot\theta\)

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

Answer:

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