Problem 36 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arccos \left(-\frac{1}{3}\right) $$

Step-by-Step Solution

Answer

The result is approximately 109.47 degrees when rounding to two decimal places.

Step 1: Input the Expression into the Calculator

Turn on your calculator and select the 'inverse cosine' or 'arccos' function. Then input -1/3. Make sure to type the '-1/3' as a fraction, not a decimal.

Step 2: Calculate

After input, press the equals button '=' to get the calculation result.

Step 3: Rounding the Result

When the result appears on the calculator screen, it may have many decimal places. Use your calculator's 'round' function to round the result to two decimal places.

Key Concepts

Arccosine Function

The arccosine function, often denoted as \( \arccos(x) \), is the inverse of the cosine function. It calculates the angle whose cosine is a given number. This function helps us determine angles when provided with the cosine value in trigonometry. In mathematical terms, if \( \cos(\theta) = x \), then \( \theta = \arccos(x) \), where \( \theta \) is in the range of \([0, \pi]\) radians. This is because the cosine function decreases from 1 to -1 as the angle increases from 0 to \( \pi \) radians.

When working with the arccosine function, it is essential to remember that it outputs angles. Therefore, when you input \(-\frac{1}{3}\) as in this exercise, \( \arccos \left(-\frac{1}{3}\right) \) will yield an angle in radians (or degrees, depending on the calculator settings).

Understanding the arccosine function is crucial in trigonometry since it allows for solving triangles and analyzing periodic functions. It is helpful not only in pure math but also in physics and engineering applications.

Trigonometry Calculators

Trigonometry calculators are tools designed to simplify calculations involving trigonometric functions such as sine, cosine, tangent, and their inverses like arccosine. These calculators are equipped with functions that handle angles, providing results in degrees or radians, depending on your preference. Using a calculator for the arccosine function involves selecting the 'inverse cosine' or 'arccos' mode. Then, you input the value you wish to evaluate, such as \(-\frac{1}{3}\).

Most calculators today are digital and allow for intuitive navigation. They often include additional functionalities like handling fractions and performing rounding, ensuring accurate results even with the most complex expressions.

Tips for using trigonometry calculators:

• Ensure the calculator is set to the correct mode (degrees or radians) based on the problem requirements.
• Familiarize yourself with the calculator's interface to quickly access functions like arccos.
• Check if your calculator directly handles fractions to maintain precision.
• Look for a built-in rounding feature to simplify final adjustments to results.

Rounding Decimal Places

Rounding decimals is a fundamental skill in mathematics to simplify numbers and focus on the most significant figures. In the context of the arccosine function, after getting the decimal approximation from a calculator, one might need to round it to make it more readable or fit the required precision level.

When rounding to two decimal places, you must look at the third decimal place:

• If it is 5 or greater, increase the second decimal place by one.
• If it is less than 5, keep the second decimal place as is.

This rounding method ensures accuracy and consistency across mathematical calculations.

Let’s consider an example: if the calculator outputs \( 1.9108 \) for an angle, rounding to two decimal places gives you \( 1.91 \). This process helps align your results with typical mathematical and scientific notation standards. It is particularly useful when reporting intermediate steps or final answers.