Problem 49 - Algebra 2

Question

Find one angle with positive measure and one angle with negative measure coterminal with each angle. \(\frac{7 \pi}{6}\)

Step-by-Step Solution

Answer

Positive: \(\frac{19\pi}{6}\), Negative: \(-\frac{5\pi}{6}\).

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial side and terminal side. They differ by integer multiples of full rotations, or in radians, by integer multiples of \(2\pi\). To find coterminal angles with a given angle, we add or subtract \(2\pi\) from it.

Step 2: Finding a Positive Measure Coterminal Angle

Since \(\frac{7\pi}{6}\) is already positive, we add \(2\pi\) to find a coterminal angle with a greater positive measure. \[\frac{7\pi}{6} + 2\pi = \frac{7\pi}{6} + \frac{12\pi}{6} = \frac{19\pi}{6} \]Thus, \(\frac{19\pi}{6}\) is a positive measure coterminal angle.

Step 3: Finding a Negative Measure Coterminal Angle

To find a coterminal angle with a negative measure, we subtract \(2\pi\) from \(\frac{7\pi}{6}\):\[\frac{7\pi}{6} - 2\pi = \frac{7\pi}{6} - \frac{12\pi}{6} = -\frac{5\pi}{6} \]This results in \(-\frac{5\pi}{6}\), which is a negative measure coterminal angle.

Key Concepts

Radians

Radians are a way to measure angles, much like degrees. However, radians follow a slightly different concept. One radian is the angle created when the arc length is equal to the radius of the circle. Radians are crucial in trigonometry and calculus because they provide a natural way to describe angles in terms of the circle's size.
Here’s why radians are useful:

• A complete circle is divided into 2π radians.
• This is because the circumference of a circle is 2π times its radius, thus making 2π radians equivalent to 360 degrees.
• Using radians allows for simple formulas, as they incorporate the circle's geometry directly.

When working with radians, always remember that to convert radians to degrees, you multiply by \(\frac{180}{\pi}\). Conversely, to convert degrees to radians, you multiply by \(\frac{\pi}{180}\). This transformation helps when you need to switch between the two measures in your calculations.

Positive Angle Measure

Positive angle measures are angles that are measured counterclockwise from the initial side. It implies starting at the positive x-axis and rotating the angle counterclockwise around the origin.
Here’s what to keep in mind:

• In radians, any angle larger than 0 is considered positive.
• To find a coterminal angle with a positive measure, add \(2\pi\) to an existing angle.
• This is because adding \(2\pi\) simply adds one whole rotation of the circle, bringing you back to the starting location.
• An example would be the angle \(\frac{19\pi}{6}\), which is coterminal with \(\frac{7\pi}{6}\) by adding \(2\pi\).

Positive angle measures are vital in defining direction and orientation in rotations, physics, and engineering.

Negative Angle Measure

When dealing with negative angle measures, the concept is fairly straightforward: these angles are measured in the clockwise direction from the initial side. A negative angle essentially reverses the standard direction of angle measurement.
Consider the following points:

• Any angle less than 0 in radians is negative.
• To determine a negative coterminal angle, you subtract \(2\pi\) from a given positive angle. This moves the terminal side clockwise, maintaining the same orientation.
• For instance, subtracting \(2\pi\) from \(\frac{7\pi}{6}\) results in \(-\frac{5\pi}{6}\), a negative-angle coterminal equivalent.

Negative angle measures are often used in computer graphics and navigation. They provide an alternative interpretation for rotational directions on a plane.