Problem 115 - Precalculus

Question

Use a calculator to evaluate the following expressions. If you get an error, explain why. $$\cot 270^{\circ}$$

Step-by-Step Solution

Answer

\( \cot 270^{\circ} \) is undefined because \( \tan 270^{\circ} \) is undefined.

Step 1: Understanding Cotangent

The cotangent function is the reciprocal of the tangent function. So, \( \cot \theta = \frac{1}{\tan \theta} \). We need to find \( \cot 270^{\circ} \).

Step 2: Calculate \( \tan 270^{\circ} \)

First, determine \( \tan 270^{\circ} \). The angle \( 270^{\circ} \) lies on the negative y-axis, where the tangent is undefined because it involves division by zero. Therefore, \( \tan 270^{\circ} \) is undefined.

Step 3: Evaluate the Reciprocal

Since \( \cot \theta = \frac{1}{\tan \theta} \), and \( \tan 270^{\circ} \) is undefined, \( \cot 270^{\circ} \) is also undefined. This is because you cannot take the reciprocal of an undefined value.

Key Concepts

Understanding Cotangent

The cotangent function, represented as \( \cot \theta \), is one of the six main trigonometric functions. It is simply the reciprocal of the tangent function. This means it is calculated as \( \cot \theta = \frac{1}{\tan \theta} \). Cotangent helps in solving problems that involve right triangles and circles.

By knowing that cotangent relies on the tangent function, understanding the behavior of tangent at specific angles is key to determining cotangent's value. Cotangent is often used in trigonometry to analyze angles and can be expressed in terms of different trigonometric identities. In right triangle terms, this is equal to the adjacent side divided by the opposite side.

What is Tangent?

The tangent of an angle, denoted as \( \tan \theta \), is a fundamental trigonometric function. It can be defined as the ratio of the opposite side to the adjacent side in a right triangle.

- For angle \( \theta \), \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \).- In a unit circle, the tangent function is the y-coordinate divided by the x-coordinate.

The value of tangent can greatly vary depending on the angle. It can be positive or negative and ranges from -∞ to +∞. When dealing with specific angles like 90°, 270°, and others, you might encounter situations where the tangent function becomes undefined.

Understanding Undefined Values

An undefined value in mathematics occurs when a mathematical operation or function doesn't yield a specific number. With trigonometric functions, undefined values happen when you attempt calculations that involve division by zero. For instance, at angles like 90° and 270°, the tangent function is undefined because it would require dividing by zero.

When a function, like the tangent, hits an undefined point, it means the function doesn't produce a clear numeric result. In trigonometry, this concept is crucial because it marks the angles at which the trigonometric ratios cannot be determined within the usual arithmetic sense.

Exploring Division by Zero

Division by zero is a situation in mathematics where a number is divided by zero. Unlike division by any other number, dividing by zero doesn't follow the normal rules because there is no number that multiply by zero will result in a non-zero number. Thus, the operation is considered operationally "undefined."

This concept is essential in understanding why some values of trigonometric functions are undefined. For example:

• The tangent function approaches division by zero in a right triangle when the angle leads to an undefined tangent value, such as 90° or 270°.
• The undefined state arises because the y-value from the unit circle is divided by an x-value of zero.

Understanding division by zero in trigonometry helps in recognizing points where computations "break down" due to this mathematical anomaly.