Problem 21 - 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. \(\tan ^{-1} 0.75\)

Step-by-Step Solution

Answer

In order to find the rounded inverse tangent of 0.75, a calculator needs to be used. After finding the inverse tangent and remembering to round to the nearest hundredth place, the approximate overall answer will determined in the given format. Since the calculation is performed using a calculator, the exact answer may slightly vary depending on the model of your calculator.

Step 1: Find the inverse tangent

Using a scientific calculator, find the inverse tangent or arctan of 0.75. This can usually be done through pressing the '2nd' or 'shift' button, then the 'tan' button, and then inputting 0.75.

Step 2: Output and Rounding

Once the arctan is found, note the value that the calculator displays. It will be a decimal number. The task asks to round this value to the nearest hundredth which is the second digit after the decimal point. In rounding, if the third digit after the decimal point is 5 or greater, you add one to the second digit. If it's less than 5, you leave the second digit as it is.

Key Concepts

Scientific Calculator Usage

When tackling mathematical problems, especially trigonometry, a scientific calculator is an indispensable tool. To compute the inverse tangent or arctan of a number, you'll need to familiarize yourself with a few basic functions of your calculator.

First, identify the button marked 'tan', which is used for calculating the tangent of an angle. Its inverse function, arctan, is usually accessed by pressing a secondary function key like '2nd' or 'shift' followed by the 'tan' button. Once you've entered the mode for arctan, you can input the number you're working with. For instance, to find the arctan of 0.75, you would enter 0.75 after pressing the sequence for the inverse function. After inputting the number, hit the 'equals' or 'enter' button to reveal the calculated value.

Some scientific calculators might have dedicated buttons for inverse functions, like 'arctan', 'arcos', or 'arcsin'. If you have such a calculator, you can directly press the 'arctan' button without the need for a secondary function key. Always consult your calculator's manual if you're unsure how to perform certain operations, and practice with different functions to become more comfortable with its capabilities.

Rounding Decimals

The concept of rounding decimals is vital when precise measurements are not necessary or when you want to simplify a number for ease of understanding. To round a decimal to the nearest hundredth, focus on the third digit after the decimal point.

If the third digit is 5 or higher, you increase the second digit by one. For example, if you have a decimal like 0.746, since 6 (the third digit) is greater than 5, you would round up to 0.75. If the third digit is less than 5, you leave the second digit unchanged. For instance, a decimal of 0.742 will be rounded down to 0.74 because the third digit, 2, is less than 5.

Here's a quick guide to help remember how to round decimals to the nearest hundredth:

• If the third decimal place is 0-4, do not change the second decimal place.
• If the third decimal place is 5-9, increase the second decimal place by 1.

Always check your answers after rounding to ensure they make sense in the context of your problem.

Understanding Arctan of a Number

The arctan function, also known as the inverse tangent, is a fundamental concept, especially when dealing with right-angled triangles in trigonometry. The arctan of a number gives you the angle whose tangent is that number.

It's used to figure out angles when you know the opposite side and the adjacent side of a right-angled triangle. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Conversely, the arctan provides the angle when given this ratio. For example, the arctan of 0.75 means you're looking for an angle whose tangent (opposite over adjacent) is 0.75.

It's important to remember that the output of an arctan function is an angle, usually measured in degrees or radians. Knowing how to calculate this can help in many practical situations, such as when determining the pitch of a roof, the angle of a hill, or adjusting the angle of a ramp. Being familiar with arctan is therefore not only key for textbook exercises but also for real-world applications where angles play a crucial role.