Understanding Co-terminal Angles: Finding the Smallest Positive Angle

Introduction

When dealing with angles in mathematics and trigonometry, co-terminal angles are pairs of angles that have the same terminal side but differ by a multiple of 360 degrees. This concept is crucial in various applications, including graphic design, engineering, and physics. One common task is to find the smallest positive co-terminal angle for a given negative angle. This article explains the process with a detailed example.

Problem Statement

The problem we address here is to find the smallest positive angle co-terminal with -1125 degrees.

Step-by-Step Solution

To find the smallest positive angle co-terminal with -1125 degrees, follow these steps:

Step 1: Division by 360

The first method is to divide -1125 by 360 to find how many full circles fit into -1125 degrees and then add 360 degrees the appropriate number of times to get a positive angle.

-1125 ÷ 360  -3.125

This indicates that -1125 degrees is equivalent to -4 full circles when rounded down to the nearest whole number.

Step 2: Calculate the Equivalent Angle

Now, to find the equivalent angle:

-1125   360 × 4  315°

Therefore, the smallest positive angle co-terminal with -1125 degrees is 315 degrees.

Using a Custom Formula

Another method to solve this involves the use of a specific formula:

Formula: f(D) 90 * INT(MOD(D, 360) / 90) * SIGN(MOD(D, 90)) - 89 * INT(MOD(D, 360) / 90) * SIGN(MOD(D, 90))

Substituting D -1125 into the formula:

f(-1125)  4

The smallest positive angle in the fourth quadrant is 271 degrees, or 270 degrees plus an additional degree.

Direct Calculation Method

You can also calculate the smallest positive angle by finding the remainder when -1125 is divided by 360 and then adding 360 degrees:

-1125 ÷ 360 gives a remainder of -45-45   360  315°

Alternatively, you can keep adding 360 degrees to -1125 until you get a positive angle:

-1125   360  -765-765   360  -405-405   360  -45-45   360  315°

This confirms that 315 degrees is indeed the smallest positive co-terminal angle with -1125 degrees.

Conclusion

Determining the smallest positive co-terminal angle is a valuable skill in mathematical and scientific contexts. Whether you use a direct calculation method, the division method, or a custom formula, the result is the same. The smallest positive angle co-terminal with -1125 degrees is 315 degrees.

Understanding how to manipulate and interpret angles is critical in many areas of study, making this knowledge indispensable for students and professionals alike.