Problem 42 - Calculus: One and Several Variables

Question

Evaluate to four decimal place accuracy. $$\cos (-13.461)$$.

Step-by-Step Solution

Answer

After evaluating \(\cos (-13.461)\) and rounding off, the final result can be obtained. An exact answer here is not provided because the output of a trigonometric function is dependent on how the calculator is set (whether radians or degrees).

Step 1: Evaluate the cosine

Use the calculator to find the cosine of -13.461. You can input this as \(\cos (-13.461)\)

Step 2: Round-off the result

After you have found the cosine using the calculator, next is to round off your answer to four decimal places. Ensure the fifth decimal number is considered in rounding off. If the fifth decimal is 5 or more the fourth decimal place is increased by one.

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry. It is one of the primary trigonometric functions used to relate the angles and sides of a right triangle. In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse.
If you imagine a triangle, the hypotenuse is the longest side, and the adjacent side is the side closest to a given angle (excluding the hypotenuse). Thus, if \(\theta\) represents an angle, then the cosine is mathematically represented as:

• \(\cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\)

The cosine function is periodic with a period of \(2\pi\), which means it repeats its values every \(2\pi\) radians. It is also an even function, so \(\cos(\theta) = \cos(-\theta)\). This feature makes calculations involving negative angles straightforward, as seen in the original exercise where \(\cos(-13.461)\) is evaluated.
Understanding how the cosine function behaves is crucial within mathematics, as it helps in solving numerous geometrical problems and in analyzing periodic phenomena in physics and engineering.

Rounding Numbers

Rounding numbers is a common practice in mathematics to simplify numbers to a specific degree of accuracy. When calculating a trigonometric function, the result is often a long decimal number. In many cases, it is practical to round this number to make calculations easier to understand and communicate.
To round a number to four decimal places, you look at the fifth decimal place:

• If the fifth number is 5 or greater, you increase the fourth decimal place by one.
• If the fifth number is less than 5, you leave the fourth decimal as it is.

For example, if the result of a calculation is 0.123456, rounding to four decimal places would yield 0.1235, because the fifth decimal (5) prompts an increase in the fourth decimal place.
Rounding is not just a mathematical formality; it affects precision and accuracy, making it an important skill in science, statistics, and everyday life.

Calculations Using Calculator

Using a calculator for trigonometric calculations simplifies finding values like \(\cos(-13.461)\). Here’s a basic guide on performing these steps using a scientific calculator:

• Ensure your calculator is in the correct mode: radians or degrees, depending on your problem's requirements. For cosine, it’s often necessary to use radians.
• Input the angle into the calculator. For the given exercise, you would type \(-13.461\), then press the "cos" function button.
• Many calculators provide results to multiple decimal places. Review the entire number before considering rounding it.
• Remember, calculators might sometimes require resetting or verification when outputs seem unusual.

By mastering your calculator's functions, you manage more complex mathematical tasks accurately. In modern education, being fluent with these tools enhances understanding and practical application of mathematical theories.