How Do You Find the Arctangent of 1.5?

Calculating the arctangent, also known as the inverse tangent, of a value such as 1.5 can be approached in several ways. In this guide, we’ll delve into how to find arctan(1.5), including the use of calculators, mathematical logic, and practical methods.

Using a Scientific Calculator

The arctangent function, often denoted as arctan or tan^{-1}, can be evaluated using a scientific calculator or mathematical software. To find arctan(1.5), follow these steps:

Input the Value: Enter 1.5 on your calculator. Select the Arctangent Function: Locate the arctan or tan^{-1} button. On some calculators, you might need to press the 2nd or shift key to activate this function. Obtain the Result: The calculator will display the result in radians or degrees, depending on the mode you had set. The result is approximately:

arctan(1.5) ≈ 0.9828 radians ≈ 56.31°

Detailed Calculation Steps

Here’s a detailed breakdown of the calculation:

Step 1: Use a Calculator
> Enter 1.5
> Press the Trigonometic Button, 2nd, and then tan^-1.
> The result is 0.9828 radians or 56.31°. Step 2: Convert to Degrees and Minutes
> 56°:18 minutes:35.8 seconds
> This can be done by breaking down the decimal part into minutes and seconds.

Understanding the Result Through Trigonometry

To better understand how this result is derived, consider the following:

1. **Using a Right Triangle**
– If you set up a right triangle with a perpendicular of 1.5 (x 1) and a base of 1, you can calculate the hypotenuse using the Pythagorean theorem: hypotenuse √(1^2 1.5^2) ≈ 1.8028. The angle between the base and the hypotenuse can be found using a protractor, which gives you approximately 56.31°.

2. **Clarks Tables for Reference**
- According to Clarks tables (a historical reference table of trigonometric values), tan(56°18') is approximately 1.5, confirming the result.

The Range of the Arctangent Function

To better understand the range of the arctan function, consider the following:

Understanding the Range: The arctangent function returns an angle whose tangent is the input value. For arctan(1.5), the angle is between 45° and 60° because tan(45°) 1 and tan(60°) ≈ 1.732. Practical Angle Range: In the first quadrant, as the angle increases from 45° to 60°, the tangent value increases.

By verifying the angle with a right triangle and checking the tangent values at 45° and 60°, you can confirm that the angle arctan(1.5) lies between 53° and 60°.

Conclusion

Calculating arctan(1.5) can be done using a scientific calculator, logical reasoning with right triangles, and historical trigonometric reference tables. The result, approximately 0.9828 radians or 56.31°, can be confirmed through these methods.

For a comprehensive understanding of arctangent and related trigonometric functions, consider using online resources, textbooks, or specialized software tools.