Problem 5

Question

For each function value, write the value or tell why it is undefined. Do not use a calculator. $\tan \frac{\pi}{2}$

Step-by-Step Solution

Verified

Answer

The value of $ \tan \frac{\pi}{2} $ is undefined because it results in division by zero.

1Step 1: Understanding Tangent

The tangent function, denoted as $ \tan $, is defined as the ratio $ \frac{\sin \theta}{\cos \theta} $. To find the value of $ \tan \frac{\pi}{2} $, we need to understand the behavior of sine and cosine at this angle.

2Step 2: Sine Value at $ \frac{\pi}{2} $

The sine of $ \frac{\pi}{2} $ is 1, because $ \sin \frac{\pi}{2} $ corresponds to the y-coordinate of the point on the unit circle at $ 90^\circ $, where $ (0,1) $ is the coordinate.

3Step 3: Cosine Value at $ \frac{\pi}{2} $

The cosine of $ \frac{\pi}{2} $ is 0, because $ \cos \frac{\pi}{2} $ is the x-coordinate of the point on the unit circle at $ 90^\circ $, which is 0.

4Step 4: Division by Zero in Tangent

Substituting these values into the tangent function, we get $ \tan \frac{\pi}{2} = \frac{1}{0} $. Division by zero is undefined in mathematics, thus $ \tan \frac{\pi}{2} $ is undefined.

Key Concepts

The Unit CircleSine and Cosine ValuesWhy Tangent is an Undefined Function at $ \frac{\pi}{2} $The Concept of Division by Zero

The Unit Circle

The unit circle is a brilliant tool for understanding trigonometric functions. Think of it as a circle with a radius of 1, centered at the origin of the coordinate plane.
The key advantage of the unit circle is that it allows us to visualize the relationships between angles and various trigonometric functions. Angles in the unit circle are typically measured in radians.
A full circle is equal to 2π radians or 360 degrees. As you move counterclockwise around this circle, the coordinates of the point where the terminal side of the angle meets the circle give you the cosine and sine of that angle.

The x-coordinate represents the cosine value.
The y-coordinate represents the sine value.

By using the unit circle, you can effortlessly determine these sine and cosine values for common angles, without needing a calculator.

Sine and Cosine Values

Sine and cosine are fundamental trigonometric functions that arise from the unit circle. Whenever you hear sine and cosine, think about the y and x coordinates of a circle with a radius of 1.
For sine, you look at the y-axis, while for cosine, it's the x-axis you focus on. Let's explore the sine and cosine values at $ \frac{\pi}{2} $ radians.

At $ \frac{\pi}{2} $ (or 90 degrees), the point on the unit circle is at $(0, 1)$.
This means that the sine of $ \frac{\pi}{2} $ is 1, since the y-coordinate is 1.
The cosine of $ \frac{\pi}{2} $ is 0 because the x-coordinate is 0.

Understanding these values is crucial for solving problems involving trigonometric functions, especially when combined like in the tangent function.

Why Tangent is an Undefined Function at $ \frac{\pi}{2} $

The tangent function, represented as $ \tan \theta $, is defined as the ratio of sine to cosine: $ \frac{\sin \theta}{\cos \theta} $. To understand why the tangent of $ \frac{\pi}{2} $ is undefined, we need to look at these sine and cosine values.

At $ \frac{\pi}{2} $, the sine value is 1, and the cosine value is 0.
Putting these into the tangent function gives $ \tan \frac{\pi}{2} = \frac{1}{0} $.

Since division by zero is undefined in mathematics, $ \tan \frac{\pi}{2} $ is likewise undefined. This is why, on graphs of the tangent function, you'll often see vertical asymptotes, or gaps, at these points.

The Concept of Division by Zero

Division by zero is a cornerstone concept in mathematics. It's one of those peculiar things that you simply cannot do. Attempting to divide any number by zero doesn't result in an infinite number or a finite value but rather something that is undefined.
This stems from the fact that dividing means determining how many times the divisor fits into the dividend.

If the divisor is zero, it cannot fit into any number of the dividend parts, leading to ambiguity.
Because zero times any number is still zero, there's no way to say how many times it fits, so it's undefined.

Understanding that division by zero is undefined helps keep our mathematical world consistent and avoids contradictions that could disrupt calculations and concepts, like with the tangent function at $ \frac{\pi}{2} $.

Problem 5