Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Give an expression that generates all angles coterminal with each angle. Let \(n\) represent any integer. $$-\frac{3 \pi}{4}$$

Step-by-Step Solution

Answer

The expression is \(-\frac{3\pi}{4} + 2\pi n\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side. To find angles coterminal with a given angle, we add or subtract multiples of a full circle (in radians, that's \(2\pi\)).

Step 2: Identifying Key Values

The given angle is \(-\frac{3\pi}{4}\). A full rotation in radians is \(2\pi\). We need to determine the expression that includes this angle plus any integer multiple of \(2\pi\).

Step 3: Formulating the Expression

To find coterminal angles, we take the given angle \(-\frac{3\pi}{4}\) and add \(2\pi n\), where \(n\) is any integer. This gives the expression \(-\frac{3\pi}{4} + 2\pi n\).

Step 4: Verification

To verify, substitute different integer values for \(n\) (e.g., 1, -1, 2) and ensure each resulting angle still maintains the essence of the original angle by being coterminal.

Key Concepts

Radians

Radians are a unit of angle measurement commonly used in mathematics. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of the circle itself. One full circle is equal to, or approximately 6.283 radians, which is noted as \[ 2\pi \] Radians provide a clear and direct link between the arc length and the radius, which can be more intuitive when dealing with circular motion and periodic functions.

• 1 radian equals approximately 57.3 degrees.
• A half-circle is \( \pi \) radians or approximately 180 degrees.
• Trigonometric functions like sine and cosine often use radians for their calculations.

Understanding radians is crucial for topics like coterminal angles because it helps in easily conveying the concept of adding or subtracting circles (or \( 2\pi \)) to find related angles.

Full Circle

A full circle in radians is represented by \( 2\pi \). This measurement corresponds to 360 degrees. The idea of a full circle is central to finding coterminal angles, as these angles share a terminal side after completing complete rotations around the circle.Here are a few key points about full circles in radians:

• When you add or subtract \( 2\pi \) radians, the angle remains essentially unchanged in position, resulting in coterminal angles.
• The concept of a full circle is helpful not only in geometry but also in physics for describing oscillations and rotations.
• The use of full circles helps simplify complex angle calculations.

Understanding the full circle allows us to predict how angles behave when we continuously rotate around a circle.

Integer Multiples

The notion of integer multiples is important when discussing coterminal angles (and other periodic phenomena). Integer multiples allow us to determine how many times a basic unit, like a full circle in angle measurement, is added or subtracted.In the context of the problem:

• We use the expression \(-\frac{3\pi}{4} + 2\pi n\) where \( n \) is an integer to calculate coterminal angles.
• Here, \( n \) represents how many full circle rotations are applied to the original angle \(-\frac{3\pi}{4}\)
• The integer \( n \) can be any number (positive, negative, or zero), letting us find an infinite number of angles.

Using integer multiples provides a systematic way of exploring the infinite possibilities of coterminal angles.

Angle Measurement

Angle measurement is key to understanding and finding coterminal angles, as it helps us define directions and rotations. Both radians and degrees are used in different contexts to measure angles, but often radians are preferred for calculations involving periodic functions.For the given angle \(-\frac{3\pi}{4}\):

• We express this angle not only in radians but also in terms of its coterminal counterparts by adding or subtracting full circles.
• The expression \(-\frac{3\pi}{4} + 2\pi n\) illustrates how we can find other coterminal angles by varying \( n \).

Accurate angle measurement allows us to understand positional relationships in trigonometry, geometry, and even real-world applications like engineering and navigation.