Problem 37 - Precalculus: A Concise Course

Question

Use a calculator to evaluate the expression. Round your result to two decimal places. $$ \tan ^{-1} \frac{19}{4} $$

Step-by-Step Solution

Answer

The rounded answer will be the value that you get after rounding the result from Step 3 to two decimal places.

Step 1: Express the Number as a Decimal

The number inside the arctan function is 19/4. When 19 is divided by 4, the result becomes decimal form. So, first, carry out the division to convert the fraction into decimal form.

Step 2: Use the atan Function

Now, use a calculator to calculate the arctan of the resulting decimal number from Step 1. Enter the decimal number into the calculator's atan function. This will give the result in radians.

Step 3: Round the Result

Now that you have the result from the atan function, round this result to two decimal places as per the requirements of the exercise. Ensure you round off correctly.

Key Concepts

Understanding Arctan

Arctan, or inverse tangent, is a function that finds the angle whose tangent is a given number. It's widely used in geometry and trigonometry when you know the ratio of the opposite side to the adjacent side of a right triangle and wish to find the angle. The expression \(\tan^{-1}\frac{19}{4}\) represents the arctan of the fraction 19/4. Using a calculator, arctan can be determined, providing an angle in radians that corresponds to that ratio. It's essential to make sure your calculator is set to the correct mode (radians or degrees) before performing the operation.

Radians - A Measure of Angles

Radians are a unit of angular measure used in many areas of mathematics, especially in trigonometry. One complete revolution around a circle is equivalent to \(2\pi\) radians, which is approximately 6.28319. Unlike degrees, which are based on dividing a circle into 360 arbitrary units, radians provide a direct relationship between the length of an arc of a circle and its radius. For example, an arc with a length equal to the radius of the circle subtends an angle of 1 radian.

From Degrees to Radians

When converting degrees to radians, you multiply by \(\frac{\pi}{180}\) and vice versa. Understanding radians is crucial for correctly interpreting angles when using the arctan function or any trigonometric operation.

Decimal Conversion

Decimal conversion is an essential mathematical skill that involves changing fractions into decimal numbers, which facilitate easier computation like the one needed for the arctan function. When converting a fraction, like 19/4, you divide the numerator (19) by the denominator (4). It's important to perform this step accurately to ensure that the following trigonometry calculations are based on the correct values. Many mathematical operations require numbers in decimal form because it simplifies multiplication, division, and utilization in functions like arctan.

Using Decimal in Calculations

In trigonometry, using decimal form numbers helps in obtaining more precise results especially when the inputs for functions like sine, cosine, and tangent are mostly real numbers that are easy to enter into a calculator or computer.

Rounding Numbers

Rounding numbers is a technique used to reduce the digits in a number while maintaining its value near to the original. In many practical situations, such as engineering calculations or statistical analysis, exact numbers are unnecessary and cumbersome; thus, numbers are rounded to a certain number of decimal places for simplicity. Rounding the result of \(\tan^{-1}\frac{19}{4}\) to two decimal places means to keep only two digits after the decimal point. Be careful to follow the standard rules for rounding: if the next digit is five or more, round up, and if it's less than five, round down.

Importance in Numerical Analysis

Rounding is particularly important in numerical analysis since it ensures that the results of calculations are both accurate and practical for real-world application. This allows you to effectively communicate your findings without overcomplicating with too many decimal places.