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 } \\)

Explanation:

Step1: Recall the definitions of trigonometric functions

On the unit circle, for an angle \(\theta\), if the terminal - side of the angle intersects the unit circle at the point \((x,y)\), then \(\sin\theta=y\), \(\cos\theta = x\), \(\tan\theta=\frac{y}{x}(x
eq0)\), \(\csc\theta=\frac{1}{y}(y
eq0)\), \(\sec\theta=\frac{1}{x}(x
eq0)\), \(\cot\theta=\frac{x}{y}(y
eq0)\).
When \(\theta = 0^{\circ}\), the terminal - side of the angle \(\theta = 0^{\circ}\) intersects the unit circle at the point \((1,0)\).

Step2: Calculate \(\sin\theta\)

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

Step3: Calculate \(\csc\theta\)

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

Step4: Calculate \(\cos\theta\)

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

Step5: Calculate \(\sec\theta\)

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

Step6: Calculate \(\tan\theta\)

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

Step7: Calculate \(\cot\theta\)

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

Answer:

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