Problem 114 - Precalculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We need to evaluate the trigonometric function \( \tan \left(270^{\circ}\right) \). We will use a calculator to determine its value.

Step 2: Setting Up the Calculation

Input \( 270^{\circ} \) into the calculator as the angle for the tangent function. Ensure the calculator is set to degree mode, as the angle is given in degrees.

Step 3: Computing the Tangent

Try calculating \( \tan \left(270^{\circ}\right) \) by executing the function with the calculator. If an error occurs, observe the error message or result.

Step 4: Interpreting the Error

An error occurs because the tangent of \( 270^{\circ} \) is undefined. The angle \( 270^{\circ} \) corresponds to an angle on the negative y-axis, where the cosine of the angle is 0, making the tangent undefined (since \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)).

Key Concepts

Tangent Function Explained

The tangent function, abbreviated as "tan," is one of the basic trigonometric functions. It relates the angle of a right triangle to the ratio of the triangle's opposite side to its adjacent side. Mathematically, the tangent of an angle \( \theta \) is expressed as:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

For angles on the unit circle, the tangent function can also be represented as the sine of the angle divided by the cosine of the angle:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

The tangent function is periodic, repeating its values every 180 degrees or \( \pi \) radians. It is important to understand that tangent values can range from negative infinity to positive infinity, which makes it slightly different from sine and cosine functions, which are always between -1 and +1.

Understanding Undefined Values

When dealing with trigonometric functions, particularly tangent, you may encounter situations where the function is undefined. This occurs when the denominator in the tangent's definition is zero. Specifically, the tangent function becomes undefined when cosine is zero:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \rightarrow \text{undefined when } \cos(\theta) = 0 \)

At angles such as 90 degrees, 270 degrees, or any odd multiple of 90 degrees, the cosine value is zero, leading to division by zero. In mathematical terms, dividing by zero results in an undefined value, which is why tangent yields an error in such scenarios.
This type of undefined nature is crucial for understanding periodic function behavior and when calculating certain angles in trigonometry.

Angle Measurement Basics

Angle measurement is fundamental in geometry and trigonometry. There are different units for measuring angles, but the most commonly used are degrees and radians. Degrees divide a full circle into 360 parts, where a right angle is 90 degrees. Radians measure angles based on the circumference of a circle, with one full rotation equating to \( 2\pi \) radians.
When calculating trigonometric functions like tangent, it is critical to know in which unit the angle is measured. This ensures accurate computation and interpretation. For example, specifying an angle as 270 may imply degrees or radians, and the outcomes differ greatly depending on the context assumed.

Working with Degree Mode

A calculator can be set to different modes to accommodate angle measurements in either degrees or radians. Setting the calculator to the correct mode is essential for accurate trigonometric calculations. For example, when working on problems involving angles in degrees, like \( 270^{\circ} \), it is important to set your calculator to degree mode.

• This ensures that the computation reflects the proper unit and avoids errors in results.

Setting the calculator incorrectly can lead to unexpected outcomes or errors, as the computations will assume a different angle measure. Always double-check your mode before inputting your angle to get the correct results. This step is often overlooked, yet it is pivotal to ensure precise trigonometric evaluations.