Problem 41 - Precalculus

Question

Use a calculator to evaluate the trigonometric functions for the indicated angle values. Round your answers to four decimal places. $$\tan \left(\frac{\pi}{8}\right)$$

Step-by-Step Solution

Answer

The value of \( \tan(\frac{\pi}{8}) \) is approximately 0.4142.

Step 1: Understanding the Expression

We need to evaluate the tangent of the angle \( \frac{\pi}{8} \). This angle is in radians, which is a common way to measure angles in trigonometry.

Step 2: Converting Radians to Degrees

To better understand the angle, we can convert it to degrees. Since \( \pi \) radians equal 180 degrees, \( \frac{\pi}{8} \) radians equals \( \frac{180}{8} = 22.5 \) degrees.

Step 3: Using the Calculator for Tangent

Enter \( 22.5 \) degrees into your calculator and use the tangent function to find \( \tan(22.5^{\circ}) \). Make sure your calculator is in degree mode.

Step 4: Rounding the Result

Once you have the output from the calculator, round the result to four decimal places to get a more precise value.

Key Concepts

Tangent Function

The tangent function is one of the basic trigonometric functions, alongside sine and cosine. It is usually represented as \( \tan(\theta) \) and relates an angle \( \theta \) of a right triangle to the ratio of the length of the opposite side to the adjacent side. In mathematical terms, it is expressed as:

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

This function is periodic, meaning it repeats its values in a regular pattern. Specifically, it has a period of \( \pi \) radians or 180 degrees.
Another vital point is that the tangent function is undefined for angles where the cosine component is zero, such as \( 90^{\circ} \) or \( \frac{\pi}{2} \) radians. Keeping your calculator in the correct mode (degrees or radians) is crucial when evaluating the tangent function.

Angle Conversion

Converting angles from radians to degrees, or vice versa, is an essential skill in trigonometry, especially when dealing with problems that provide angles in one measurement but require solutions in another. Here’s the conversion principle:

• 1 radian = 180/\( \pi \) degrees
• \( 1^{\circ} = \pi/180 \) radians

For example, to convert \( \frac{\pi}{8} \) radians to degrees, you use the formula:

• \( \frac{\pi}{8} \times \frac{180}{\pi} = 22.5^{\circ} \)

Remembering this conversion will be especially useful when using calculators that operate only in degree mode. It ensures that when you input an angle, your calculation is accurate.

Radian Measure

Radians are another method to express angles, favored in many mathematical contexts because of their direct relationship with circle geometry. In a circle, one radian corresponds to the angle created when the arc's length is equal to the circle's radius.
Using radians has added benefits, such as simplifying formulas in calculus, particularly those involving derivatives and integrals of trigonometric functions. It's essential to become comfortable with radians when engaging with advanced mathematics, as they are frequently used.
To understand radian measure better, remember:

• A full circle is \( 2\pi \) radians, which corresponds to 360 degrees.
• Half a circle is \( \pi \) radians or 180 degrees, and so forth.

Considering radians as a natural measure of angles can streamline how you approach and solve trigonometric problems.