Problem 41 - Precalculus : Building Concepts and Connections

Question

Use a scientific calculator to evaluate the trigonometric functions. Make sure the calculator is in DEGREE mode. Round to four decimal places. $$\sec 15^{\circ}$$

Step-by-Step Solution

Answer

The secant of 15 degrees, rounded to four decimal places, is approximately 1.0353.

Step 1: Set Calculator to Degree Mode

Before starting the calculation, it is important to check that the calculator is set to 'DEGREE' mode. This will ensure that the angles entered will be interpreted as degrees, not radians.

Step 2: Calculate Secant Function

If the calculator has the secant ('sec') function, use it directly by typing '15' and pressing the 'sec' button. If not, remember that sec x equals \(1/ \cos x\), so you could also evaluate the cosine of 15 degrees and then find its reciprocal.

Step 3: Round Off the Answer

Upon obtaining the result, round it off to four decimal places. Make sure to follow standard rounding rules. If the fifth decimal is 5 or higher, round up; if it's less than 5, round down.

Key Concepts

Secant Function

The secant function is a fundamental part of trigonometry, often denoted as \(\sec(x)\). It is one of the six primary trigonometric functions and is defined as the reciprocal of the cosine function. In other words, \[\sec(x) = \frac{1}{\cos(x)}\] This means that if you know the value of the cosine of an angle, you can easily find the secant by taking the reciprocal. The secant function is particularly important in various fields such as physics, engineering, and computer graphics.

Secant functions are used in solving problems involving lengths of sides in right triangles, among others. It is crucial to understand the behavior of the secant function, especially where it becomes undefined. Since the cosine of 90 degrees and 270 degrees (among other angles) is zero, the secant function will be undefined at these angles. Make sure to use a scientific calculator effectively when working with secant functions to ensure accurate results.

Degree Mode

Degree mode in a calculator refers to a setting where angles are measured in degrees rather than radians or gradians. This is important because trigonometric functions like cosine, sine, and secant vary based on angle measurements. If you are performing calculations in degree mode, ensure your calculator is set correctly.

When using a scientific calculator, switching to degree mode can often be done by navigating the settings or by pressing a specific mode button. Always confirm the mode before beginning calculations to avoid incorrect answers. Degrees are commonly used in geometry, geography, and many practical applications like navigation, making this mode essential for basic calculations in these fields.

Rounding Decimal Places

Rounding decimal places is a critical step in calculations, especially when precision is required. After performing trigonometric calculations like finding the secant of an angle, rounding the result ensures clarity and accuracy. The task might involve rounding off to a set number of decimal places, for instance, four decimal places as stated in the original exercise.

To round a number to four decimal places, consider the fifth decimal. If it is 5 or greater, the fourth decimal is rounded up. If it is less than 5, it remains the same. This follows general rounding rules but is particularly important in scientific settings where exactness matters. Using a calculator efficiently to manage rounding ensures that your math results remain precise and communicable across various applied contexts.