Problem 15 - Technical Mathematics with Calculus

Question

Evaluate the following, giving your answer in decimal degrees to three significant digits. $$\arctan 4.26$$

Step-by-Step Solution

Answer

The angle in decimal degrees is approximately 77.145 degrees, rounded to three significant digits.

Step 1: Understanding the Arctan Function

Arctan, or inverse tangent, is the function that returns the angle whose tangent is a given number. In this case, we need to find the angle whose tangent is 4.26. To do so, we can use a calculator with trigonometric function capabilities.

Step 2: Using a Calculator

Set your calculator to degree mode. Input the value 4.26 and apply the arctan (or tan⁻¹) function to it. This will provide you with the angle in degrees that corresponds to the tangent of 4.26. Ensure that you round the answer to three significant digits.

Step 3: Reporting the Answer

Once you have the angle from the calculator, write down the angle in decimal degrees, rounded to three significant digits. This is the final answer.

Key Concepts

Tangent of an Angle

The tangent of an angle in a right-angled triangle is a fundamental concept in trigonometry. It is defined as the ratio of the length of the opposite side to the length of the adjacent side. To put it simply, if you have a right-angled triangle and you're looking at one of the non-right angles, the tangent of that angle is found by dividing the triangle’s opposite side by its adjacent side.

For an angle \( \theta \) in a right triangle, we express this relationship as \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \). What makes the tangent particularly interesting is its ability to relate an angle to a ratio of sides, allowing us to solve for one if we know the other. This is especially useful in various fields like engineering, physics, and especially in navigation, where it helps in calculating distances that are not directly measurable.

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the action of the basic trigonometric functions like sine, cosine, and, pertinent to our exercise, tangent. When we have the ratio of sides, but we need to find the measure of an angle, we use these inverse functions.

The inverse of the tangent function is known as \( \arctan \) or \( \tan^{-1} \) and it gives us the angle whose tangent is a particular number. To visualize this, imagine we have a known slope of a line, and we want to find the angle it makes with the x-axis; \( \arctan \) helps us find this angle. It’s important to remember that because of the periodic nature of the tangent function, \( \arctan \) will only give us angles in the range of \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) radians, or from -90 to 90 degrees.

Significant Digits

Significant digits, or significant figures, are a way of expressing precision in a number. They include all digits in a number that contribute to its accuracy, starting with the first non-zero digit and ending with the last reliably known digit. When numbers are used in calculations, like using the \( \arctan \) function on a calculator, it’s crucial to know how many significant digits are appropriate to keep.

For example, if a result from a calculator shows 81.869897 but we want to express this with three significant digits, we round this to 81.9. The concept of significant digits ensures that the precision of the number reflects the precision of the measurement or calculation, which prevents overestimating the accuracy of a result.

Decimal Degrees

When angles are measured in degrees, we often use a decimal form to express them, known as decimal degrees. This is in contrast to the more traditional degrees, minutes, and seconds (DMS) format. Decimal degrees simplify many mathematical calculations since they can be treated just like any other numerical value in calculations.

In our exercise, after using the \( \arctan \) function, we would find the angle measure in decimal degrees. Then, we round this as per the required significant digits. It's essential to be familiar with converting and using decimal degrees as many scientific calculators and software default to this format when working with angles.