Problem 28 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a calculator to approximate the value of the expression, if possible. Round your answer to the nearest hundredth. \(\tan ^{-1} 5.9\)

Step-by-Step Solution

Answer

The value of \(\tan^{-1} 5.9\) rounded to the nearest hundredth is approximately 80.54 degrees.

Step 1: Input Value

The value given to find the inverse tangent of is 5.9. Hence, we are trying to solve for \(\tan^{-1} 5.9\)

Step 2: Use Calculator

Use a calculator to calculate the inverse tangent of the given value. Make sure your calculator is in degree mode because the default mode for most calculators is radians. After ensuring this, input tan^(-1)5.9 and press equals or calculate to get the result.

Step 3: Round to the Nearest Hundredth

Once you have the result from the calculator, round it to the nearest hundredth. To round to the nearest hundredth, look at the digit in the thousandths place. If this digit is five or greater, increase the digit in the hundredths place by one. If it is less than five, leave the digit in the hundredths place as it is. The rest of the digits following should be dropped.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in mathematics that help us find angles when given trigonometric ratios. For instance, the inverse tangent, denoted as \(\tan^{-1}\), allows us to determine the angle whose tangent is a specific number. This is also known as the arctangent function.

When you see \(\tan^{-1} 5.9\), it means you are looking for an angle whose tangent equals 5.9. This function is crucial in various fields like engineering and physics, where understanding angles is necessary.

• Inverse functions are often useful for solving triangles, converting between angle measures, and in calculus.
• They primarily return values in specific ranges. For \(\tan^{-1}\), this range is usually from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\) (or equivalently \(-90^\circ\) to \(90^\circ\) in degrees).

Understanding inverse trigonometric functions can significantly enhance your problem-solving skills by allowing you to reverse-engineer trigonometric calculations.

Use of Calculator in Mathematics

Calculators are versatile tools in mathematics that simplify complex calculations and provide precise results quickly. When calculating inverse trigonometric functions like \(\tan^{-1} 5.9\), a calculator can make the process more manageable.

Here’s how you typically use a calculator for this purpose:

• Ensure that your calculator is set to the correct mode. Most calculators have both degree and radian modes. For most basic applications, including this problem, you need to be in degree mode.
• Input the inverse tangent function by finding the \((\tan^{-1})\) button on your calculator. Enter the value provided, in this case, 5.9.
• Press equals or calculate to receive your result.

This method helps avoid manual errors and saves time. However, understanding the underlying concepts is just as crucial as using the calculator itself, as it helps you better interpret the results.

Rounding Numbers

Rounding numbers is a straightforward concept in mathematics that makes numbers simpler and easier to work with, especially in approximations. In our step-by-step solution, after we use the calculator to find \(\tan^{-1} 5.9\), the result might have more decimal places than we need. That’s where rounding comes in.

To round to the nearest hundredth:

• Identify the digit in the hundredths place.
• Look at the digit in the thousandths place right after it.
• If the thousandths digit is 5 or higher, increase the hundredths digit by one.
• If it’s less than 5, leave the hundredths digit as it is and drop the rest of the digits.

Rounding ensures that your final answer is both precise and easy to communicate, especially in instances of scientific notation or when presenting your solution in a classroom setting.