Problem 126 - Precalculus

Question

For the following exercises, use a calculator to evaluate each expression. Express answers to the nearest hundredth. $$ \tan ^{-1}(6) $$

Step-by-Step Solution

Answer

The value of \(\tan^{-1}(6)\) is approximately 1.41 radians.

Step 1: Identify the Target Function

The exercise requires evaluating the inverse tangent function \( an^{-1}(6)\). This function represents the angle whose tangent is 6.

Step 2: Use the Calculator

Turn on your calculator and make sure it is set to the right mode (degree or radian) aligned with the requirements of the problem, generally radians unless specified otherwise.

Step 3: Evaluate the Expression

Enter \( an^{-1}(6)\) into the calculator. Calculators typically have a specific key for the inverse tangent function. Once entered, observe and record the result displayed.

Step 4: Round the Result

The calculator will provide the angle as a decimal. Round this value to the nearest hundredth. This is important for accuracy as required by the problem.

Key Concepts

Evaluating Expressions

Evaluating expressions, especially those involving inverse trigonometric functions, is about identifying the meaning behind those symbols. In this exercise, we look at the expression \( \tan^{-1}(6) \). This notation indicates that we want to find an angle whose tangent is 6.
Inverse trigonometric functions like \( \tan^{-1}(x) \) are used when we have the value of a trigonometric function and need to find the corresponding angle. Various calculators can handle these calculations quite efficiently. Most of them will have a button for \( \tan^{-1} \), allowing you to directly input the value and receive the corresponding angle.
To evaluate an expression properly, always break it down into:

• Identifying the function: Understand what \( \tan^{-1} \) represents here (an angle with a tangent value of 6).
• Using a calculator: Ensure it is set correctly to calculate this.

By following these steps, you simplify the task of working with seemingly complex expressions.

Radian Mode

When evaluating trigonometric expressions on a calculator, choosing between degree and radian mode is crucial for accurate results. Radian mode is often used in mathematical calculations because it's the standard unit in calculus and most scientific applications.
Radians measure angles differently compared to degrees: one full revolution is \( 2\pi \) radians instead of 360 degrees. When a problem does not specify the unit, it's safe to assume that radians should be used.
Checking your calculator's mode is simple:

• Look for a mode button, often labeled as "MODE," or similar.
• Switch to "RAD" or "radian" mode if it's not already set.

Using radian mode ensures that your calculations, including \( \tan^{-1}(6) \), align with most academic and scientific practices.

Rounding Numbers

Rounding numbers is a fundamental part of mathematics that helps express results in a more digestible form. When your calculator provides a decimal, such as when calculating \( \tan^{-1}(6) \), you may end up with a long series of digits. Rounding helps to simplify this.
For this exercise, we round to the nearest hundredth. Here's a simple guide:

• Identify the hundredths place – it's the second digit to the right of the decimal point.
• Check the digit next to it (the thousandths place) to decide whether to round up or stay the same.
• If the thousandths digit is 5 or higher, round the hundredths digit up by one. Otherwise, keep it the same.

Rounding is key for clear communication and precision, ensuring that your answer is both accurate and concise.