To find the exact value of [tex]\(\tan 270^{\circ}\)[/tex], let's break down the steps and understand the situation geometrically and mathematically.
1. Understanding the Angle on the Unit Circle:
- The angle [tex]\(270^{\circ}\)[/tex] is measured clockwise from the positive x-axis.
- Starting from [tex]\(0^{\circ}\)[/tex] at the positive x-axis and rotating 270 degrees clockwise, you will end up on the negative y-axis.
- Thus, the terminal side of [tex]\(270^{\circ}\)[/tex] lies along the line extending downward through the origin, crossing the negative y-axis.
2. Coordinate on the Unit Circle:
- When an angle lands on the y-axis at [tex]\(270^{\circ}\)[/tex], it corresponds to the point [tex]\((0, -1)\)[/tex] on the unit circle.
3. Definition of Tangent in Terms of Sine and Cosine:
- Recall that the tangent function [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex].
- For the coordinates [tex]\((0, -1)\)[/tex]:
- [tex]\(\sin 270^{\circ} = -1\)[/tex]
- [tex]\(\cos 270^{\circ} = 0\)[/tex]
4. Division by Zero:
- When you attempt to find [tex]\(\tan 270^{\circ}\)[/tex], you need to calculate [tex]\(\frac{\sin 270^{\circ}}{\cos 270^{\circ}} = \frac{-1}{0}\)[/tex].
- Division by zero is undefined in mathematics. Therefore, [tex]\(\tan 270^{\circ}\)[/tex] does not have a finite value.
5. Conclusion:
- Since the tangent function at [tex]\(270^{\circ}\)[/tex] involves division by zero, we classify the result as "undefined."
Thus, the exact value of [tex]\(\tan 270^{\circ}\)[/tex] is:
```
undefined
```
