Why Tan 90 Is Undefined

Why tan 90° is Undefined: A Deep Dive into Trigonometric Functions

The question of why the tangent of 90 degrees (tan 90°) is undefined is a fundamental concept in trigonometry that often stumps students. Understanding this requires a solid grasp of the definitions of trigonometric functions, specifically tangent, and how they relate to the unit circle. This article will delve into the intricacies of this concept, providing a comprehensive explanation accessible to anyone with a basic understanding of trigonometry. We will explore the unit circle, the definition of tangent, and the implications of approaching 90° from different angles.

Understanding Trigonometric Functions and the Unit Circle

Trigonometric functions, such as sine, cosine, and tangent, are defined in relation to a right-angled triangle. However, a more powerful and general way to understand them is through the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) of a coordinate plane.

Any point on the unit circle can be represented by its coordinates (x, y). If we draw a line from the origin to this point, forming an angle θ (theta) with the positive x-axis, we can define the trigonometric functions as follows:

• Cosine (cos θ): The x-coordinate of the point on the unit circle.
• Sine (sin θ): The y-coordinate of the point on the unit circle.
• Tangent (tan θ): The ratio of the y-coordinate to the x-coordinate: tan θ = sin θ / cos θ = y/x

This unit circle definition allows us to extend the domain of trigonometric functions beyond the angles found in right-angled triangles, allowing us to consider angles greater than 90° and even negative angles.

Visualizing the Problem: Approaching 90° on the Unit Circle

Let's consider what happens as we approach an angle of 90° on the unit circle. As the angle θ approaches 90°, the point on the unit circle moves closer and closer to the point (0, 1).

• Cosine (cos θ): As θ approaches 90°, the x-coordinate (cos θ) approaches 0.
• Sine (sin θ): As θ approaches 90°, the y-coordinate (sin θ) approaches 1.
• Tangent (tan θ): Since tan θ = sin θ / cos θ, as θ approaches 90°, we get a situation where the numerator approaches 1 and the denominator approaches 0.

This is where the problem arises. Dividing by zero is undefined in mathematics. No matter how close θ gets to 90°, we are always dividing by a number extremely close to, but not equal to, zero. The result of this division becomes infinitely large, approaching positive infinity.

The Mathematical Explanation: Limits and Infinity

We can use the concept of limits in calculus to express this more formally. The limit of a function f(x) as x approaches a value 'a' is written as:

lim (x→a) f(x)

In our case, we want to find the limit of tan θ as θ approaches 90°:

lim (θ→90°) tan θ

As θ approaches 90° from below (i.e., from angles slightly less than 90°), tan θ approaches positive infinity (+∞). If we approach 90° from above (i.e., from angles slightly greater than 90°), we would find that tan θ approaches negative infinity (-∞). Since the limit from the left and the right are not equal, the limit itself is undefined.

Why is it Important to Understand this?

Understanding why tan 90° is undefined is crucial for several reasons:

• Foundation of Trigonometry: This concept forms a fundamental building block for more advanced trigonometric concepts and calculations.
• Calculus and Analysis: The idea of limits and undefined values is central to calculus and mathematical analysis. Understanding this helps to build a solid foundation for these advanced topics.
• Applications in Physics and Engineering: Trigonometric functions are extensively used in physics and engineering to model various phenomena, such as waves, oscillations, and projectile motion. Understanding the limitations of these functions is crucial for accurate modeling and analysis.
• Avoiding Errors in Calculations: Recognizing that tan 90° is undefined prevents errors in calculations and ensures that results are accurate and meaningful.

Exploring Tangent's Behavior Near 90° through Examples

Let's illustrate the behavior of the tangent function as we approach 90° using specific examples:

• 89°: tan 89° ≈ 57.29
• 89.9°: tan 89.9° ≈ 572.96
• 89.99°: tan 89.99° ≈ 5729.58
• 89.999°: tan 89.999° ≈ 57295.78

As you can see, as the angle gets closer to 90°, the value of the tangent function increases dramatically. The same pattern is observed when approaching 90° from the other side (e.g., 91°, 90.1°, etc.), but with negative values.

Addressing Common Misconceptions

A common misconception is that tan 90° is equal to infinity (∞). While the tangent function approaches infinity as the angle approaches 90°, infinity is not a number in the traditional sense. It represents a concept of unbounded growth. Therefore, tan 90° is not equal to infinity but rather undefined. The distinction is important because you cannot perform arithmetic operations with infinity in the same way you would with real numbers.

Frequently Asked Questions (FAQ)

Q: What about tan 270°?

A: Similar to tan 90°, tan 270° is also undefined. As the angle approaches 270° on the unit circle, the cosine approaches 0, leading to division by zero in the tangent function.

Q: Can we use a calculator to find tan 90°?

A: Most calculators will return an error message (e.g., "Error," "Undefined," or "Math Error") when you try to calculate tan 90°. This is because the calculator recognizes that the calculation is invalid.

Q: Are there any other angles where the tangent function is undefined?

A: The tangent function is undefined at any angle where the cosine is zero. This occurs at odd multiples of 90°, i.e., 90°, 270°, 450°, etc., and their negative counterparts.

Conclusion: A Crucial Concept in Mathematics

The undefinition of tan 90° is not merely a mathematical quirk; it's a fundamental concept that underscores the importance of understanding the underlying principles of trigonometric functions and the limitations of mathematical operations. By grasping this concept, we solidify our foundation in trigonometry, preparing us for more complex mathematical concepts and real-world applications. Remember, understanding the behavior of trigonometric functions near points of discontinuity is vital for accurate calculations and a deeper appreciation of the elegance and power of mathematics. The fact that tan 90° is undefined highlights the importance of careful mathematical reasoning and the need to always consider the limitations of our mathematical tools.