Problem 34 - Algebra and Trigonometry

Question

\(31-36\) me 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 \(\frac{32\pi}{3}\) and \(\frac{11\pi}{3}\) are not coterminal.

Step 1: Define Coterminal Angles

Coterminal angles are angles that share the same terminal side when in standard position. They differ by a multiple of full rotations, which is equivalent to a multiple of \(2\pi\) radians.

Step 2: Calculate Difference of Angles

Subtract one angle from the other to see if the resulting difference is a multiple of \(2\pi\).\[\frac{32\pi}{3} - \frac{11\pi}{3} = \frac{32\pi - 11\pi}{3} = \frac{21\pi}{3} = 7\pi\]

Step 3: Check for Multiples of Full Rotations

Check if the difference \(7\pi\) is a multiple of \(2\pi\). Since \(7\pi\) divided by \(2\pi\) is not an integer, the angles are not multiples of a full rotation.

Key Concepts

Angle Subtraction

Understanding how to subtract angles is crucial when determining if angles are coterminal. In this exercise, we are given two angles, \( \frac{32\pi}{3} \) and \( \frac{11\pi}{3} \), expressed in radians. To find out if these angles are coterminal, we subtract one angle from the other:

• Subtract the smaller angle from the larger angle.
• The subtraction is \( \frac{32\pi}{3} - \frac{11\pi}{3} \).
• Simplifying gives \( \frac{21\pi}{3} \), which reduces to \( 7\pi \).

Determining the difference allows us to see if it's a multiple of \( 2\pi \), which is necessary for understanding coterminality. Angle subtraction helps simplify and compare complex angle measures.

Standard Position

An angle is said to be in standard position when its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis. This position serves as the reference for measuring angles:

• The angle rotates counterclockwise to form a positive angle.
• If the angle rotates clockwise, it forms a negative angle.

Placing angles in standard position makes it easier to compare because they share a common starting point. In the standard position, it is straightforward to visualize how far an angle has rotated. Thus, it helps to determine whether two angles share the same terminal side.

Multiples of Rotations

Determining if two angles are coterminal involves finding if their difference is a multiple of a full rotation. A full rotation is \( 2\pi \) radians.

• If an angle's rotation is a whole number multiple of \( 2\pi \), then it ends at the same point on the circle as another angle.
• For our example, the difference \( 7\pi \) must be checked against \( 2\pi \).
• Calculating, we have \( \frac{7\pi}{2\pi} = 3.5 \), which is not an integer.

The non-integer result means the angles are not coterminal. Understanding multiples of rotation clarifies how angles can return to the same terminal position on a circle.