Understanding Division by Zero and the Concept of tan 90 Degrees

In mathematics, particularly in calculus, division by zero and the concept of tan 90 degrees often raise questions regarding mathematical definitions and limits. This article aims to clarify these concepts and explain why division by zero and tan 90 degrees are both undefined, using examples and detailed explanations.

Division by Zero

Let us first discuss the concept of division by zero. In mathematics, dividing any number by zero is undefined. This is because, mathematically speaking, division involves the inverse of multiplication. For example:

[Mathematical Expression]

However, the product of any number and zero is zero. Therefore, we cannot determine a single value that, when multiplied by zero, equals a non-zero number. This is the fundamental reason why division by zero is undefined.

tan 90 Degrees

The tangent function, denoted as tan, is a ratio of the sine and cosine functions. Specifically, tan x sin x / cos x. At x 90 degrees (or π/2 radians), the value of cos x is zero. Thus, we have:

[Mathematical Expression]

This expression results in division by zero, which is undefined. Additionally, as x approaches 90 degrees, the value of tan x grows without bound. This is why we say that tan 90 degrees is infinity, but it is not a defined number in the context of tan function.

Equating Division by Zero to tan 90 Degrees

It is often incorrectly stated that any number divided by zero equals tan 90 degrees. This statement is not accurate because both are undefined. To clarify, consider the following example:

x/0 undefined

Here, x can be any real number. However, this expression is not mathematically valid and thus cannot be equated to tan 90 degrees, which is also undefined. Both concepts fall under the same category of undefined in mathematics.

Graphical Representation

To further understand why division by zero and tan 90 degrees are both undefined, let us consider their graphical representations. Let us draw a circle with a radius of ten units and plot the tangent function on this circle:

Step 1: Draw a Circle

On your graph paper, draw a circle with a radius of 10 units, ensuring your graph paper is at least 20 units wide and tall. Label the axes from 0 at the center by tenths until you reach 1.0 where the circle crosses each axis.

Step 2: Plot Tangent on the Circle

The tangent of an angle is the y-coordinate divided by the x-coordinate at the point where the line for the angle crosses the circle. However, at 90 degrees, there is no point on the circle where the x-coordinate is zero and the y-coordinate is non-zero, which is necessary to compute tan 90 degrees.

Conclusion

From these graphical and mathematical explanations, it is clear that both division by zero and tan 90 degrees are undefined concepts. Attempting to equate one to the other is incorrect and can lead to mathematical misunderstandings. Therefore, it is crucial to recognize the undefined nature of these concepts in mathematical contexts.