Problem 46 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the giren trigonometric functions to four decimal places. $$\tan (-1)$$

Step-by-Step Solution

Answer

The value of \( \tan(-1) \) to four decimal places, as calculated on a scientific calculator in radian mode, is -1.5574

Step 1: Understand the Tangent Function

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. The key to evaluating the tangent of any angle, including negative angles, is to realize that the tangent function is periodic, with period \( \pi \). Thus, the tangent of an angle \( \theta + \pi \) is the same as the tangent of \( \theta \). For negative angles, this means the tangent of a negative angle is equivalent to the tangent of that angle added to \( \pi \).

Step 2: Convert to Positive Angle if Necessary

As īndicated in Step 1, the tangent of negative one is the same as the tangent of \( \pi -1 \), so we need to find the tangent of \( \pi - 1 \)

Step 3: Use a Scientific Calculator

Input the value we got from step 2 ('\( \pi - 1 \)') in the scientific calculator. Ensure that the calculator is set to 'radian' mode, as the tangent function typically assumes that input values are in radians. Now press the '= ' button to get the precise value to four decimal places.

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions related to right triangles. In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

It is part of the basic SOHCAHTOA trigonometric mnemonic, specifically:

• Tangent (tan): Opposite / Adjacent

This function is important in analyzing triangles and modeling periodic phenomena.

One noteworthy property of the tangent function is its periodicity. Tangent has a period of \( \pi \) radians, meaning that \( \tan(\theta) = \tan(\theta + n\pi) \) for any integer \( n \). This property helps in evaluating the function for negative angles, as negative angles can be converted to their equivalent positive angle by adding \( \pi \).

For instance, \( \tan(-1) \) can be converted to \( \tan(\pi - 1) \), leveraging the periodic nature of the tangent function.

Scientific Calculator Usage

A scientific calculator is a critical tool in evaluating trigonometric functions accurately. Using it requires understanding how to switch modes between degrees and radians, as trigonometric functions can take arguments in both measurements. In trigonometry, radians are often preferred for mathematical functions.

When performing calculations such as \( \tan(\pi - 1) \), you must ensure that your calculator is set to radian mode. Here are the steps you generally need to follow to use a scientific calculator for such tasks:

• Power on your scientific calculator.
• Find and press the "mode" button to change the setting to 'radians'.
• Input the expression you wish to evaluate, such as \( \pi - 1 \).
• Press the function key "tan" before entering the expression, or select it as per calculator design.
• Compute the expression by pressing "=" to get the result.

Using these steps, you should obtain the tangent value to the desired precision.

Radians

Radians are a unit of angular measure used in many areas of mathematics. They are based on the radius of a circle, making them a natural way to describe angles.

When considering a complete circle, it sums up to \( 2\pi \) radians, equivalent to 360 degrees. This equivalency allows conversion between radians and degrees using the formula:

• \( \text{Degrees} = \frac{\text{Radians} \times 180}{\pi} \)
• \( \text{Radians} = \frac{\text{Degrees} \times \pi}{180} \)

Utilizing radians is particularly significant in various trigonometric calculations since most calculators and mathematical functions default to radian mode. This is because radians provide a direct proportion to the arc length: a one radian angle cuts off an arc whose length is equal to the radius. Furthermore, using radians instead of degrees simplifies the differentiation and integration of trigonometric functions in calculus due to their derivations aligning more seamlessly with the periodicity and characteristics of these functions.

Thus, understanding radians is crucial for solving trigonometric exercises accurately.