Question

evaluate the six trigonometric function of $\theta$. (hint: sketch the angle and use your unit circle) if the ratio is undefined, enter dne.
$\theta = 540^{\circ}$
$\

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

$

Explanation:

Step1: Find the coterminal angle

Since \(540^{\circ}-360^{\circ} = 180^{\circ}\), \(\theta = 540^{\circ}\) is coterminal with \(180^{\circ}\).
On the unit - circle, for an angle \(\theta\) in standard position, if \(\theta = 180^{\circ}\), the coordinates of the point on the unit - circle are \((x,y)=(- 1,0)\).
The definitions of the trigonometric functions are:
\(\sin\theta=y\), \(\cos\theta=x\), \(\tan\theta=\frac{y}{x}(x
eq0)\), \(\csc\theta=\frac{1}{y}(y
eq0)\), \(\sec\theta=\frac{1}{x}(x
eq0)\), \(\cot\theta=\frac{x}{y}(y
eq0)\)

Step2: Calculate \(\sin\theta\)

Using the definition \(\sin\theta = y\), when \(x=-1,y = 0\), \(\sin(540^{\circ})=\sin(180^{\circ})=0\)

Step3: Calculate \(\csc\theta\)

Using the definition \(\csc\theta=\frac{1}{y}\), since \(y = 0\), \(\csc(540^{\circ})=\csc(180^{\circ})\) is undefined (DNE) because division by zero is not allowed.

Step4: Calculate \(\cos\theta\)

Using the definition \(\cos\theta=x\), when \(x=-1,y = 0\), \(\cos(540^{\circ})=\cos(180^{\circ})=-1\)

Step5: Calculate \(\sec\theta\)

Using the definition \(\sec\theta=\frac{1}{x}\), when \(x=-1\), \(\sec(540^{\circ})=\sec(180^{\circ})=\frac{1}{-1}=-1\)

Step6: Calculate \(\tan\theta\)

Using the definition \(\tan\theta=\frac{y}{x}\), when \(x=-1,y = 0\), \(\tan(540^{\circ})=\tan(180^{\circ})=\frac{0}{-1}=0\)

Step7: Calculate \(\cot\theta\)

Using the definition \(\cot\theta=\frac{x}{y}\), since \(y = 0\), \(\cot(540^{\circ})=\cot(180^{\circ})\) is undefined (DNE) because division by zero is not allowed.

Answer:

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