Problem 36 - Algebra and Trigonometry

Question

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ \frac{32 \pi}{3}, \frac{11 \pi}{3} $$

Step-by-Step Solution

Answer

The angles are not coterminal.

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. This can be identified if the difference between two angles is an integer multiple of a full circle, which in radians is \(2\pi n\), where \(n\) is an integer.

Step 2: Convert Angles to Comparable Form

We first compare the given angles \( \frac{32\pi}{3} \) and \( \frac{11\pi}{3} \). Let's first find their difference: \( \frac{32\pi}{3} - \frac{11\pi}{3} = \frac{21\pi}{3} = 7\pi \).

Step 3: Check for Integer Multiple of Circle

To check if the angles are coterminal, we need to see if the difference \(7\pi\) is an integer multiple of \(2\pi\). We divide \(7\pi\) by \(2\pi\): \( \frac{7\pi}{2\pi} = 3.5 \).

Step 4: Evaluate Integer Multiple

Since \(3.5\) is not an integer, the difference \(7\pi\) is not an integer multiple of \(2\pi\). Thus, \( \frac{32\pi}{3} \) and \( \frac{11\pi}{3} \) are not coterminal angles.

Key Concepts

Standard Position Angles

An angle in standard position is one where its vertex is at the origin of a coordinate plane, and the initial side lies along the positive x-axis. Think of this as starting from a perfectly horizontal line on the right side and measuring the angle in a counterclockwise direction. This setup is a reference point for measuring other angles, and it ensures consistency in how angles are drawn and understood.

When you picture angles in standard position, you're using a convenient method that allows easy comparison with other angles. It’s like having a universal starting point. Whether you rotate your angle slightly or make an entire loop, the starting side remains fixed. This is crucial for determining if angles are coterminal, meaning if they eventually land on the same position.

Radians

Radians are an essential way to measure angles, particularly in higher mathematics and physics. Unlike degrees, radians relate directly to the circumference of a circle. One radian is the angle created when the arc length is equal to the radius of the circle. With radians, a full circle is equated to the constant \(2\pi\), which is approximately 6.28318.

This unit of angular measurement simplifies many mathematical calculations, especially those involving calculus. In practice, knowing angles in radians allows smooth transitions between arc lengths, areas, and various trigonometric functions. For instance, the angles discussed in the exercise, \(\frac{32\pi}{3}\) and \(\frac{11\pi}{3}\), represent angular distances in radians, showcasing how distances can be expressed in circular measures rather than just degree angles.

• Degrees revolve around a circle being \(360\) degrees.
• Radians relate aspects of a circle's geometry directly.

Integer Multiples of 2pi

In the realm of circular motion and angles, integer multiples of \(2\pi\) represent full rotations. This is because \(2\pi\) radians correspond to completing an entire circle. An angle that is a multiple of \(2\pi\), whether \(2\pi, 4\pi, 6\pi\), and so forth, implies the path retraces the circle completely one or more times.

Determining if two angles are coterminal involves checking the difference between them. If this difference is an integer multiple of \(2\pi\), then they are coterminal, meaning they terminate at the same position on the circle. In the exercise, we calculated the difference \(7\pi\), and found it wasn't a simple integer multiple of \(2\pi\) because \(\frac{7\pi}{2\pi} = 3.5\), which is not a whole number.

This concept highlights how rotations can return to the same terminal side, even if the magnitude of rotation is different. Understanding integer multiples of \(2\pi\) is essential in identifying coterminal angles, an important concept when exploring periodicity and wave-like functions in trigonometry.