Problem 125 - Precalculus

Question

For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth. $$ \cos ^{-1}(0.8) $$

Step-by-Step Solution

Answer

\( \cos^{-1}(0.8) \approx 0.64 \) radians.

Step 1: Understand the Problem

We need to find the angle whose cosine is 0.8. This is the value of the inverse cosine function, referred to as \( \cos^{-1}(0.8) \).

Step 2: Use a Calculator

Using a scientific calculator, input the value 0.8 into the inverse cosine function. Ensure the calculator is set to degrees or radians based on the requirement, but here we'll find the default result as radians.

Step 3: Convert the Result

If the calculator provides the result in radians and you need degrees, you would convert it. However, since the question does not specify, we typically assume radians unless indicated otherwise.

Step 4: Round to Nearest Hundredth

Once you have the result from the calculator, look at the third digit after the decimal point to determine rounding. If it is 5 or more, round up.

Key Concepts

Understanding the Cosine Function

The cosine function is a fundamental concept in trigonometry. It is one of the three main functions used to relate angles to the ratios of sides in right triangles, the others being sine and tangent. The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

• In general terms, for a triangle with an angle \( \theta \), cosine is defined as \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \).
• This function is often used when you need to find angles given side lengths.
• The inverse cosine function, written as \( \cos^{-1} \), allows you to determine an angle when the cosine value is known. This is crucial when solving trigonometric equations.

When you see \( \cos^{-1}(0.8) \), it means you are looking for an angle whose cosine is 0.8. This number can be found using a calculator, often yielding a result in radians.

Exploring Radians

Radians are an alternative way to measure angles, unlike the more familiar degrees. Understanding radians is vital for working effectively in trigonometry and calculus.

• One full circle is equivalent to \( 2\pi \) radians, which is roughly 6.283 radians.
• A right angle is \( \frac{\pi}{2} \) radians or about 1.57 radians.
• Radians provide a natural way of describing angles when working with mathematical functions, because they are ratios derived from a circle's properties.

In many cases, calculators set to default return angle measures in radians rather than degrees. Unless otherwise specified, it is common to assume radians in mathematical contexts. However, converting between the two can be done when necessary: \[\text{Degrees} = \text{Radians} \times \left( \frac{180}{\pi} \right)\]This conversion ensures you can navigate between different measurement systems with ease.

The Importance of Rounding Numbers

In mathematical solutions, rounding numbers helps in simplifying the values and making them easier to interpret. It's especially important when results are needed with a certain precision, such as to the nearest hundredth.

• Rounding involves evaluating the digit following the desired precision. Here, rounding to the nearest hundredth means focusing on the third decimal place.
• If this third digit is 5 or more, you round up the second digit; if it is less, the value stays.
• For example, given a result of 0.823, rounding to the nearest hundredth gives 0.82; but for 0.826, it rounds to 0.83.

Accurate rounding minimizes errors in calculations and helps maintain consistency in reporting numerical data. Double-check your rounding to ensure clarity and precision in your answers.

Utilizing a Scientific Calculator

A scientific calculator is a powerful tool for solving mathematical problems, especially those involving functions like cosine and its inverse. These calculators offer a range of functions and settings that can drastically simplify mathematic operations.

• Ensure the calculator is in the correct mode (degrees or radians) based on your requirements. Since many problems imply radians, this mode is often the default.
• Enter the value into the inverse cosine function to find the angle. For example, with \( \cos^{-1}(0.8) \), directly use the button for \( \cos^{-1} \) and input 0.8.
• Scientific calculators make it easy to verify answers by allowing rapid recalculations. They also often store previous results, so you can adjust inputs as necessary without starting over.

Using these functions effectively requires familiarity, so take some time to explore the different capabilities of your scientific calculator. Practice makes perfect when it comes to efficiently using this essential tool.