What Is The Tan Of 90

The tangent function, a cornerstone of trigonometry, reveals the intricate relationships between the angles and sides of right triangles. It helps us understand the steepness of lines and curves, and is fundamental in fields ranging from physics to engineering. But what happens when we ask a seemingly simple question: what is the tan of 90 degrees? This seemingly straightforward inquiry opens the door to fascinating mathematical concepts and challenges our intuitive understanding of trigonometric functions.

Defining the Tangent Function

Before diving into the specifics of tan 90, it's crucial to understand the fundamental definition of the tangent function. In a right triangle, the tangent of an angle (θ) is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle. Mathematically, this is expressed as:

tan(θ) = Opposite / Adjacent

This definition works perfectly well for angles within the range of 0 to 90 degrees (excluding 90 degrees itself) within a right triangle. However, when we start exploring angles beyond this range, we need a more general definition using the unit circle.

The Unit Circle and the Tangent Function

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Any point on the unit circle can be defined by its coordinates (x, y), which also correspond to (cos θ, sin θ), where θ is the angle formed by the positive x-axis and the line connecting the origin to the point.

In this context, the tangent function can be redefined as:

tan(θ) = sin(θ) / cos(θ)

This definition is equivalent to the opposite/adjacent definition for angles within a right triangle, but it extends the concept of the tangent function to any angle, positive or negative, and even beyond 360 degrees.

Understanding 90 Degrees

An angle of 90 degrees, also known as a right angle, is a quarter of a full rotation. On the unit circle, a 90-degree angle corresponds to the point (0, 1). This means that:

• cos(90°) = 0
• sin(90°) = 1

The Dilemma of Tan 90

Now, let's apply the unit circle definition of the tangent function to find tan(90°):

tan(90°) = sin(90°) / cos(90°) = 1 / 0

Herein lies the problem: division by zero. In mathematics, division by zero is undefined. It doesn't result in a specific number; rather, it leads to a concept of infinity or undefinedness.

Why Division by Zero is Undefined

Division can be thought of as the inverse operation of multiplication. When we say 10 / 2 = 5, it's because 2 * 5 = 10. So, if tan(90°) = x, then 0 * x would have to equal 1. However, no matter what value we assign to x, 0 multiplied by x will always be 0, never 1. This fundamental incompatibility is why division by zero is undefined.

Approaching 90 Degrees: A Limit Approach

While tan(90°) is undefined, we can explore what happens to the tangent function as we approach 90 degrees from values slightly less than 90 degrees.

Consider an angle θ that is very close to 90 degrees, say 89.9 degrees, 89.99 degrees, or even 89.999 degrees. As θ gets closer and closer to 90 degrees, sin(θ) approaches 1, and cos(θ) approaches 0. This means that tan(θ) = sin(θ) / cos(θ) becomes a very large number.

Mathematically, we express this using the concept of a limit:

lim (θ→90-) tan(θ) = +∞

This reads as "the limit of tan(θ) as θ approaches 90 degrees from the left (values less than 90) is positive infinity."

Approaching 90 Degrees from the Other Side

We can also approach 90 degrees from values slightly greater than 90 degrees, such as 90.1 degrees, 90.01 degrees, or 90.001 degrees. In this case, sin(θ) still approaches 1, but cos(θ) becomes a very small negative number. Therefore, tan(θ) becomes a very large negative number.

Mathematically, this is expressed as:

lim (θ→90+) tan(θ) = -∞

This reads as "the limit of tan(θ) as θ approaches 90 degrees from the right (values greater than 90) is negative infinity."

The Tangent Function's Behavior Around 90 Degrees

The fact that the tangent function approaches positive infinity from the left and negative infinity from the right around 90 degrees is a critical piece of information. It demonstrates that the tangent function has a vertical asymptote at 90 degrees.

A vertical asymptote is a vertical line that a function approaches but never touches. In the case of tan(θ), the line θ = 90 degrees is a vertical asymptote. As θ gets closer to 90 degrees, the value of tan(θ) shoots off towards either positive or negative infinity.

Implications and Applications

The undefined nature of tan(90°) and the behavior of the tangent function around 90 degrees have significant implications in various fields:

• Calculus: In calculus, understanding limits and asymptotes is crucial for analyzing the behavior of functions. The tangent function serves as a prime example of a function with a vertical asymptote and how to deal with it using limit concepts.
• Engineering: In engineering, especially in fields like surveying and navigation, trigonometric functions are used extensively. Knowing that tan(90°) is undefined helps engineers avoid potential errors when dealing with angles approaching 90 degrees.
• Physics: In physics, particularly in optics and wave mechanics, the tangent function is used to describe angles of incidence, reflection, and refraction. Understanding its behavior at and around 90 degrees is essential for accurate calculations.
• Computer Graphics: In computer graphics, trigonometric functions are used for rotations, projections, and other transformations. The behavior of the tangent function needs to be carefully considered when dealing with near-vertical lines or planes.

Practical Examples and Visualization

To further illustrate the concept, consider these practical examples:

• The Leaning Ladder: Imagine a ladder leaning against a wall. The angle between the ladder and the ground is θ. As the ladder becomes more vertical (θ approaches 90 degrees), the height the ladder reaches on the wall (opposite side) increases dramatically for a small change in the distance from the wall (adjacent side). The ratio of the height to the distance, which is the tangent of the angle, becomes increasingly large. At 90 degrees, the ladder is perfectly vertical, and the distance from the wall is zero, making the tangent undefined.
• The Searchlight: Imagine a searchlight shining a beam of light onto a distant wall. The angle of the searchlight is θ. As the searchlight points more and more directly upwards (θ approaches 90 degrees), the point where the beam hits the wall moves further and further away. In theory, as the angle reaches 90 degrees, the beam would have to travel an infinite distance to hit the wall, illustrating the concept of the tangent approaching infinity.

Common Misconceptions

There are a few common misconceptions related to tan(90°) that are worth addressing:

• Thinking Tan(90°) is Infinity: While it's true that tan(θ) approaches infinity as θ approaches 90 degrees, it's crucial to remember that infinity is not a number. Therefore, tan(90°) is not equal to infinity; it is undefined. Infinity is a concept representing unbounded growth.
• Trying to Assign a Value to Tan(90°): Some people attempt to assign a specific value to tan(90°) based on patterns or approximations. However, the fundamental principle of division by zero being undefined prevents any such assignment from being valid.

Conclusion

So, what is the tan of 90 degrees? The answer is that it is undefined. This arises from the definition of the tangent function as the ratio of sine to cosine and the fact that cos(90°) equals zero, leading to division by zero. While tan(90°) itself is undefined, understanding the behavior of the tangent function as it approaches 90 degrees – diverging towards positive and negative infinity – is essential in mathematics, science, and engineering. This exploration of tan(90°) underscores the importance of understanding limits, asymptotes, and the fundamental principles of trigonometry.

Frequently Asked Questions (FAQ)

• Why is tan(90°) undefined?

  Tan(90°) is undefined because it is equal to sin(90°) / cos(90°), which is 1 / 0. Division by zero is undefined in mathematics.

• Does tan(90°) equal infinity?

  No, tan(90°) does not equal infinity. While tan(θ) approaches infinity as θ approaches 90 degrees, infinity is not a number, and tan(90°) remains undefined.

• What is the limit of tan(θ) as θ approaches 90 degrees?

  The limit of tan(θ) as θ approaches 90 degrees from the left (values less than 90) is positive infinity (+∞). The limit of tan(θ) as θ approaches 90 degrees from the right (values greater than 90) is negative infinity (-∞).

• What is a vertical asymptote, and how does it relate to tan(90°)?

  A vertical asymptote is a vertical line that a function approaches but never touches. The line θ = 90 degrees is a vertical asymptote for the tangent function because as θ gets closer to 90 degrees, the value of tan(θ) approaches either positive or negative infinity.

• In what fields is the understanding of tan(90°) important?

  The understanding of tan(90°) is important in fields such as calculus, engineering, physics, and computer graphics, where trigonometric functions are used extensively.

• Can I calculate tan(89.99°) using a calculator?

  Yes, you can calculate tan(89.99°) using a calculator. The result will be a very large number, approximately 5729.57. This illustrates how the tangent function increases dramatically as the angle approaches 90 degrees.

• Is there any real-world situation where tan(90°) is actually encountered?

  In many practical applications, you would avoid situations where the angle is exactly 90 degrees because the tangent is undefined. However, understanding the behavior of the tangent function as it approaches 90 degrees is crucial for dealing with near-vertical angles in fields like surveying, navigation, and engineering.

• How does the unit circle help in understanding tan(90°)?

  The unit circle provides a visual and conceptual framework for understanding trigonometric functions beyond the confines of right triangles. By defining the tangent function as sin(θ) / cos(θ), where (cos θ, sin θ) are the coordinates of a point on the unit circle, it becomes clear that tan(90°) is undefined because cos(90°) = 0.

• What is the difference between undefined and indeterminate forms in mathematics?

  Undefined refers to expressions like division by zero (a/0), where the operation is not mathematically permissible. Indeterminate forms, such as 0/0 or ∞/∞, arise in the context of limits, and their value cannot be determined immediately. Further analysis, such as L'Hôpital's Rule, is required to evaluate them. Tan(90°) is undefined, not indeterminate.

• Why do calculators sometimes give an error when calculating tan(90°)?

  Calculators give an error when calculating tan(90°) because they are programmed to recognize that division by zero is undefined. The calculator cannot produce a numerical value for an undefined expression.

By addressing these FAQs, we can provide a more comprehensive understanding of the intricacies of tan(90°) and its implications in mathematics and related fields.