Problem 30 - Precalculus Essentials

Question

Use a calculator to find the value of each expression rounded to two decimal places. $$\tan ^{-1}(-\sqrt{5061})$$

Step-by-Step Solution

Answer

The answer is approximately -89.46 degrees.

Step 1: Calculate square root

First, find the square root of 5061 which is approximately 71.14.

Step 2: Find inverse tangent

Secondly, find the inverse tangent of 71.14. Many scientific calculators will have a function to compute the inverse tangent, often labeled 'tan^-1' or 'atan'.

Step 3: Negative value

After finding the inverse tangent value, multiply it by -1 to get the value of \(-\tan^{-1}(\sqrt{5061})\).

Step 4: Round off the result

Finally, round off your answer to two decimal places.

Key Concepts

Understanding Square Roots

The square root of a number is another number that, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4 because 4 times 4 equals 16. When dealing with larger numbers, calculating square roots can become cumbersome without a calculator. Staying accurate is important, especially in exercises where further calculations depend on this value.

To find the square root of a number like 5061, a calculator simplifies the process. Most calculators have a dedicated square root button, typically depicted as \( \sqrt{} \). By pressing this button and entering 5061, the approximate result, 71.14, is obtained. Note that square roots can be irrational, which means they often result in non-terminating decimals. Calculators round these to a manageable number of digits.

Proper Calculator Usage

Using a calculator might seem straightforward, but understanding what buttons to press ensures accurate outcomes. When asked to find values like \( \tan^{-1}(-\sqrt{5061}) \), specific steps and buttons are needed.

Here is a simple process to follow:

• First, input the calculation for a square root.
• Then, use the inverse trigonometric function—often labeled as 'tan^-1' or 'atan'.
• Follow the order of operations if multiple functions are involved.
• Ensure you use parentheses where needed, for example, when calculating \( -\tan^{-1}(\text{result}) \).

This ensures that each part of the mathematical operation is computed correctly, providing a reliable final answer.

Mastering Rounding Numbers

Rounding is a mathematical technique used to simplify numbers to a desired degree of precision. In many cases, such as exams or engineering, precision is crucial. Rounding helps make numbers less unwieldy while retaining as much useful information as possible.

Here's a quick guide for rounding to two decimal places:

• Identify the third decimal place.
• If this number is 5 or higher, increase the second decimal by one.
• If it's less than 5, keep the second decimal the same.

By applying this method to your calculations, you ensure that all rounded numbers are consistent and accurate, allowing for clearer communication of numerical results.

Scientific Calculator Functions

Scientific calculators are powerhouses when dealing with complex calculations, such as trigonometric functions and square roots. They often include functions like sine (sin), cosine (cos), and tangent (tan), with their inverse functions: arcsine (sin^-1), arccosine (cos^-1), and arctangent (tan^-1).

When looking to find an inverse tangent, understanding your calculator's layout is key. Here's how to get started:

• Switch your calculator to the appropriate mode (degree or radian, depending on your problem).
• Enter the input value—like the square root result—in the tan^-1 function.
• Some calculators may require pressing a '2nd' function key first.

Being familiar with these functions and the calculator's user manual allows you to efficiently solve even the most challenging mathematical problems.