Question

question
given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function.
$\tan 180^circ$
answer
$\tan 180^circ = $

Explanation:

Step1: Recall the definition of tangent on the unit circle

On the unit circle, for an angle \(\theta\), the coordinates of the point corresponding to the angle are \((\cos\theta, \sin\theta)\). The tangent function is defined as \(\tan\theta=\frac{\sin\theta}{\cos\theta}\).

Step2: Find \(\cos 180^{\circ}\) and \(\sin 180^{\circ}\)

For \(\theta = 180^{\circ}\), the point on the unit circle is \((- 1,0)\). So, \(\cos180^{\circ}=-1\) and \(\sin180^{\circ}=0\).

Step3: Calculate \(\tan 180^{\circ}\)

Using the formula \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), substitute \(\theta = 180^{\circ}\), \(\sin180^{\circ}=0\) and \(\cos180^{\circ}=-1\) into the formula:
\(\tan180^{\circ}=\frac{\sin180^{\circ}}{\cos180^{\circ}}=\frac{0}{-1}\)

Step4: Simplify the fraction

\(\frac{0}{-1} = 0\)

Answer:

\(0\)