Problem 35 - Algebra and Trigonometry

Question

The measures of two angles in standard position are given. Determine whether the angles are coterminal. $$ \frac{5 \pi}{6}, \frac{17 \pi}{6} $$

Step-by-Step Solution

Answer

Yes, the angles are coterminal because their difference is \( 2\pi \).

Step 1: Understanding Coterminal Angles

Coterminal angles share a terminal side when drawn in standard position. To determine if two angles are coterminal, subtract the smaller angle from the larger angle. If their difference is a multiple of \( 2\pi \), they are coterminal.

Step 2: Calculating the Difference

To check if \( \frac{5\pi}{6} \) and \( \frac{17\pi}{6} \) are coterminal, we need to find the difference between these two angles: \[ \frac{17\pi}{6} - \frac{5\pi}{6} = \frac{12\pi}{6} = 2\pi \]

Step 3: Checking for Multiples of \( 2\pi \)

Since the difference, \( 2\pi \), is exactly one multiple of \( 2\pi \), the two angles \( \frac{5\pi}{6} \) and \( \frac{17\pi}{6} \) are coterminal.

Key Concepts

Standard Position

When we talk about angles in standard position, we refer to a particular way of placing and visualizing angles on a coordinate plane. In this setup, the angle's vertex is located at the origin (0, 0), and its initial side extends along the positive x-axis. This configuration allows us to easily compare and combine angles. The standard position helps in understanding the rotational nature of angles and in determining properties such as trigonometric functions or whether angles are coterminal.

Here are some key points about angles in standard position:

• Vertex at the origin (0, 0)
• Initial side along the positive x-axis
• Angle measured counterclockwise is positive; clockwise is negative

Understanding this concept is crucial, as it provides a common frame of reference to assess and compute various properties of different angles.

Multiple of 2π

Angles are often discussed in terms of radians, and a full rotation around a circle is represented by the angle of 2π radians. Understanding multiples of 2π is key in discussions about coterminal angles, as angles that differ by any integral multiple of 2π radians are coterminal. This means they share the same terminal side in the standard position.

Here is what you need to know about multiples of 2π:

• An angle of 2π represents a complete circle, bringing you back to the same point
• Two angles are coterminal if their difference is a multiple of 2π
• This concept is useful for converting angles greater than 2π into more manageable, equivalent angles

Knowing how to work with and identify these multiples can streamline your understanding of rotation and angle equivalence.

Angle Measurement

Angle measurement is an essential concept in trigonometry and geometry. The common units to express angles are degrees and radians. Radians have a direct relationship with the radius and the arc length of a circle, making them particularly useful in many mathematical contexts.

Consider what happens when we measure angles in radians:

• Radian measure is the length of the arc on the circle's circumference, with the arc length equal to the radius
• One complete revolution around a circle is 2π radians
• Radians provide a natural way to handle circular movement and rotations

Understanding how angles are measured allows for better comprehension of how angles interact with each other and with the overall geometry. This is especially important in determining whether angles are coterminal, as it involves calculations using radians and multiples of 2π.