Problem 33 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \arctan 0.92 $$

Step-by-Step Solution

Answer

The rounded value of \( \arctan 0.92 \) to two decimal places is approximately 42.87 degrees.

Step 1: Use calculator to find the value

Using a scientific calculator, find the value of \( \arctan 0.92 \). Most calculators will initially give a result in radians.

Step 2: Convert radians to degrees

To convert radians to degrees, you can use the conversion factor \( \frac{180}{\pi} \). However, most calculators have a function to convert radians to degrees directly.

Step 3: Round to two decimal places

Look at the third decimal place. If it's 5 or higher, round the second decimal place up, otherwise leave it.

Key Concepts

Use of Scientific Calculator

Mastering the use of a scientific calculator is crucial for students tackling complex mathematical concepts. These calculators are designed to handle functions beyond basic arithmetic, including trigonometric functions such as the arctangent, denoted as \(\arctan\). When evaluating \(\arctan 0.92\), the calculator performs an intricate computation that considers the ratio of the opposite side to the adjacent side of a right-angled triangle.To use a scientific calculator to find \(\arctan 0.92\), locate the \(\arctan\) or \(\tan^{-1}\) button. Enter \(0.92\) and press this button. The scientific calculator will display the angle whose tangent is \(0.92\), typically in radians by default. Familiarity with your calculator's settings is important as you may need to switch between radian and degree mode depending on your requirements.

Radians to Degrees Conversion

Radians and degrees are two units for measuring angles. While radians are often used in theoretical mathematics and calculus, degrees are more commonly used in everyday situations and many practical applications. To convert between these units, the key relationship to remember is that \(180^\circ\) is equivalent to \(\pi\) radians.

Conversion Formula

To convert an angle from radians to degrees, multiply by \(\frac{180}{\pi}\). Conversely, to convert from degrees to radians, multiply by \(\frac{\pi}{180}\). Scientific calculators typically have a built-in function to convert the angle once it's in radians; this function directly applies the necessary conversion factor.For instance, after calculating \(\arctan 0.92\), if your calculator shows the result in radians, apply the degrees conversion feature or manually multiply the radian value by \(\frac{180}{\pi}\) to obtain the result in degrees.

Rounding Decimal Places

Rounding numbers to a certain number of decimal places is a common practice to simplify numbers, making them easier to work with or understand. When an exercise specifies rounding to two decimal places, it requires the number to be precise up to two numbers following the decimal point.

Rounding Rule

Look at the third decimal place—if it is 5 or higher, round the second decimal place up; if it is 4 or lower, leave the second decimal unchanged. For example, if the calculator shows the result of \(\arctan 0.92\) in degrees as \(42.809...^\circ\), rounding to two decimal places would give \(42.81^\circ\) because the third decimal place is higher than 5.Utilizing the rounding feature on a calculator can expedite this process, but it's always beneficial to understand how rounding works manually in order to verify the precision of the results.