To answer this question, let’s first review some important aspects of right triangles and the tangent function.
1. Right Triangle Basics:
- A right triangle is a triangle in which one angle is exactly 90 degrees.
- The other two angles are acute, meaning they are both greater than 0 degrees and less than 90 degrees.
2. Tangent Function:
- The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
- Mathematically, for an angle [tex]\( \theta \)[/tex], this is written as:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
3. Critical Angles:
- [tex]\(\tan(0^\circ) = 0\)[/tex]: At 0 degrees, the opposite side of the angle is 0, and the adjacent side has some positive length. Thus:
[tex]\[
\tan(0^\circ) = \frac{0}{\text{adjacent}} = 0
\][/tex]
This makes sense because as the angle approaches 0 degrees, the height (opposite side) of the triangle becomes very small, tending towards zero. Hence, the tangent, which is the ratio, will also tend towards zero.
- [tex]\(\tan(90^\circ) = \text{undefined}\)[/tex]: For a 90-degree angle, the opposite side of the angle is as long as the hypotenuse, and the adjacent side is 0. Thus, the tangent of 90 degrees involves division by zero, which is undefined. However, in some contexts like limits and certain interpretations, it can be useful to talk about the behavior of the tangent function as it approaches 90 degrees.
4. Why is it Useful?:
- Continuity: Defining tangent values at these critical points helps maintain a continuous and predictable behavior of trigonometric functions at the boundaries. It’s important for understanding the limits and for certain applications in calculus.
- Mathematical Framework: Having these definitions allows for a more complete mathematical framework for trigonometry. For example, defining [tex]\(\tan(0^\circ) = 0\)[/tex] makes it easier to understand and visualize the behavior of the tangent function as a whole.
In summary:
- The tangent of 0 degrees is 0 because at 0 degrees, the opposite side's length is 0, making the ratio of opposite to adjacent equal to 0.
- The tangent of 90 degrees is typically considered undefined in strict terms because of the division by zero, but it's useful to consider its behavior approaching 90 degrees for applications in mathematics and physics.
This understanding is critical in both theoretical and applied mathematics for maintaining consistency across various trigonometric applications.
