Problem 14 - Calculus of a Single Variable

Question

Use a calculator to approximate the value. Round your answer to two decimal places. \(\arctan (-5)\)

Step-by-Step Solution

Answer

Using these steps, the approximate value of \(\arctan(-5)\), rounded to two decimal places, should be found. The exact answer will depend on the calculator you're using.

Step 1: Calculate the arctan

Use your calculator to find the value of \(\arctan (-5)\). Most scientific calculators should have this functionality.

Step 2: Round to Two Decimal Places

After you have computed \(\arctan (-5)\), you must round this number to two decimal places. Here, note that if the third digit after the decimal is 5 or more, you should round up the second digit. If it's less than 5, leave the second digit as it is.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine the angles when the trigonometric ratios are known. These functions include arcsine (written as \(\arcsin\)), arccosine (written as \(\arccos\)), and arctangent (written as \(\arctan\)).

Particularly, the \(\arctan\) function, also known as the inverse of the tangent function, takes a ratio (in this case, the quotient of the opposite side over the adjacent side in a right-angle triangle) and returns an angle whose tangent is the given ratio. The result is typically given in radians, but it can easily be converted to degrees using a scientific calculator or by applying the appropriate conversion factor manually.

In the exercise \(\arctan (-5)\), we're finding the angle whose tangent is -5. Since tan is negative in the second and fourth quadrants of the unit circle, this angle is located in one of those quadrants.

Scientific Calculator Usage

A scientific calculator is an invaluable tool when working with complex mathematical functions, including inverse trigonometric functions. To calculate \(\arctan (-5)\), you would look for a button labeled 'TAN' with an \(\arctan\) or 'INV' option.

Here are general steps to follow:

• Switch your calculator to the appropriate mode if necessary (degree or radian).
• Enter the value -5.
• Press the \(\arctan\) or \(\tan^{-1}\) function, sometimes accessed by pressing an 'INV' or '2nd' button first.
• Press '=' to get the result.

Remember that every calculator can vary slightly, so consulting the user manual or online resources for your specific model is a good idea if you're unsure of the steps.

Rounding Decimals

Rounding decimals is the process of shortening a number to a certain number of decimal places, making it easier to read and work with. When rounding to two decimal places, as the exercise dictates, you would look at the third decimal place.

Here's how to do it:

• If the third decimal is 5 or higher, you increase the second decimal by one (this is known as rounding up).
• If the third decimal is less than 5, you leave the second decimal as it is (rounding down).

This rounding rule ensures that the results are as accurate as possible while still being concise. For \(\arctan (-5)\), after using your calculator, you would adjust the resulting value to two decimal places following these guidelines.