Problem 57 - A Graphical Approach to Precalculus with Limits

Question

Give an expression that generates all angles co terminal with each angle. Let n represent any integer. $$30^{\circ}$$

Step-by-Step Solution

Answer

The expression is \( 30° + 360°n \), where \( n \) is an integer.

Step 1: Understand Coterminal Angles

Coterminal angles are angles that share the same initial side and terminal side but may differ in the number of rotations. They can be found by adding or subtracting full rotations (360°) from the given angle.

Step 2: Identify the Base Angle

In this exercise, the base angle provided is 30°. We will use this angle to find an expression for all coterminal angles.

Step 3: Develop the Expression for Coterminal Angles

To find all angles coterminal with 30°, we need to add or subtract integer multiples of 360° to the given angle, 30°. This can be expressed as: \[ 30° + 360°n \] where \( n \) is any integer.

Step 4: Write the Final Expression

The expression \( 30° + 360°n \) generates all angles coterminal with 30°, as \( n \) can be any positive, negative, or zero integer to provide all possible angles that share terminal sides with 30°.

Key Concepts

360 degree rotation

When we talk about a 360-degree rotation, we mean a full turn around a circle. This concept is important because it helps us understand coterminal angles, which are angles that share the same terminal side.

Imagine facing north and turning in a full circle back to facing north again. You've made a full 360-degree rotation. However, you technically didn't change your facing direction even though you moved.

This is what happens with coterminal angles. Despite rotating around the circle once or more, the resultant angle points in the same direction as the initial angle. This completeness in rotation is what allows us to add or subtract multiples of 360 degrees to find other angles that have the same terminal side, or are coterminal.

Remember:

• 360 degrees = 1 full rotation
• Coterminal angles can have different measurements due to added full rotations

integer multiples

Understanding integer multiples is essential when dealing with coterminal angles. An integer is any whole number and can be positive, negative, or zero. When we talk about integer multiples, we're referring to a number that multiplies a given number by an integer.

In the context of coterminal angles, we use integer multiples of 360 degrees. By multiplying 360 by any integer, we generate new angles that, when added to or subtracted from the initial angle, create coterminal angles.

Here’s how it works:

• A positive integer multiple results in a forward rotation (clockwise for commonly used angles).
• A negative integer multiple gives a backward rotation (counterclockwise).
• Zero would mean no rotation, returning just the original angle.

As a result, the flexible nature of integer multiples allows us to generate numerous co-terminal angles.

angle expression

An angle expression is a formula that describes all possible angles that can be derived from a given angle. When dealing with coterminal angles, the goal is to create an expression that includes all angles sharing a terminal side with an initial angle.

For the given exercise, the initial angle is 30 degrees. To create an expression that accounts for all coterminal angles, you take the initial angle and add or subtract 360-degree rotations.

The general expression: \[30° + 360°n\]Here, \( n \) represents any integer, which allows the formula to adjust in multiple rotations around the circle.

By substituting different integer values into \( n \), you generate different coterminal angles. For any real-world applications, pick the \( n \) that helps simplify your requirements, whether for design, navigation, or any circular-based calculations.

• This expression provides a powerful tool to derive any angle sharing the same direction as 30 degrees.
• Experiment with various values of \( n \) to see the effect adding or subtracting rotations have.