Problem 28 - Algebra and Trigonometry

Question

\(25-30\) me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle. $$ \frac{11 \pi}{6} $$

Step-by-Step Solution

Answer

Positive: \( \frac{23 \pi}{6}, \frac{35 \pi}{6} \); Negative: \( -\frac{\pi}{6}, -\frac{13 \pi}{6} \).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position. To find coterminal angles, we add or subtract full rotations (or multiples of \( 2\pi \), in radians) from the given angle.

Step 2: Finding Two Positive Coterminal Angles

To find positive coterminal angles with \( \frac{11 \pi}{6} \), add \( 2\pi \) to the angle. Calculate: \( \frac{11 \pi}{6} + \frac{12 \pi}{6} = \frac{23 \pi}{6} \). For another positive coterminal angle, add another \( 2\pi \): \( \frac{23 \pi}{6} + \frac{12 \pi}{6} = \frac{35 \pi}{6} \).

Step 3: Finding Two Negative Coterminal Angles

To find negative coterminal angles with \( \frac{11 \pi}{6} \), subtract \( 2\pi \) from the angle. Calculate: \( \frac{11 \pi}{6} - \frac{12 \pi}{6} = -\frac{\pi}{6} \). For another negative coterminal angle, subtract \( 2\pi \) again: \( -\frac{\pi}{6} - \frac{12 \pi}{6} = -\frac{13 \pi}{6} \).

Key Concepts

Standard Position Angles

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. The importance of this arrangement is to provide a consistent frame of reference when measuring rotational movement or when comparing angles.
The terminal side of the angle is the side that has rotated from the initial side, forming the angle itself. This configuration allows for a clear visualization of how much and in which direction (clockwise or counterclockwise) the plane has rotated to form that angle.
By always using the positive x-axis as our starting point, it simplifies the process of determining the rotation and discovering coterminal angles, both positive and negative.

• Standard position means the starting line is fixed.
• It's easier to identify how far and where the terminal side lies.
• Makes comparison and calculation straightforward.

Positive Coterminal Angles

Positive coterminal angles are those that, in their measure, exceed the given angle by complete rotations of 360 degrees or, in the context of radians, by increments of \(2\pi\).
By adding multiples of \(2\pi\) to an angle, you create angles that land on the same terminal side as the given angle but have revolved one or more complete circles around.
For example, if we have \(\frac{11\pi}{6}\), which is already a positive angle, adding \(2\pi\) to this angle gives us \(\frac{23\pi}{6}\).
Adding another \(2\pi\) results in \(\frac{35\pi}{6}\). These angles are effectively the same on the circle in terms of location but differ in terms of description based on their revolutions.

• Add \(2\pi\) repeatedly for positive coterminal angles.
• Each addition results in a higher number, increasing rotations around the circle.
• Great for applications requiring angles in a greater positive range.

Negative Coterminal Angles

Negative coterminal angles are those resulting from decreasing the given angle by whole rotations. These angles are found by subtracting increments of \(2\pi\) from the given angle. This method effectively reverses the direction of the measurement clockwise.
Let's look at \(\frac{11\pi}{6}\). By subtracting \(2\pi\) (which is equivalent to \(\frac{12\pi}{6}\)), we get a negative coterminal angle of \(-\frac{\pi}{6}\).
Performing another subtraction of \(2\pi\) yields \(-\frac{13\pi}{6}\). These operations show how angles can spiral backwards into negative territory while remaining on the same terminal side as the original angle.

• Subtraction of \(2\pi\) yields negative angles.
• Useful for angles needing expression in terms of counter-clockwise measurement.
• Demonstrates full circle subtraction while keeping the terminal side constant.