Problem 62 - Precalculus with Calculus Previews

Question

Find the angle between \(-2 \pi\) and o radians that is coterminal with the angle in Problem 49 . $$ \frac{5 \pi}{6} $$

Step-by-Step Solution

Answer

The angle is $-\\frac{7\\pi}{6}$ radians.

Step 1: Understanding Coterminal Angles

Coterminal angles are angles that share the same initial and terminal sides. They differ by full rotations (multiple of \(2\pi\) radians). When looking for a coterminal angle within a specific range, such as between \(-2\pi\) and 0 radians, we adjust our angle within these limits.

Step 2: Subtract Full Rotations

To find an angle between \(-2\pi\) and 0 that is coterminal with \(\frac{5\pi}{6}\), we need to subtract full rotations (multiples of \(2\pi\)) until the angle falls within the given range. Start with the angle \(\frac{5\pi}{6}\).

Step 3: First Rotation Adjustment

Subtract \(2\pi\) from \(\frac{5\pi}{6}\): \(\frac{5\pi}{6} - 2\pi = \frac{5\pi}{6} - \frac{12\pi}{6} = -\frac{7\pi}{6}\).

Step 4: Verify Range

The result, \(-\frac{7\pi}{6}\), is greater than \(-2\pi\) and less than 0. This means the angle is within the required range of \(-2\pi\) to 0 radians.

Key Concepts

Radians

Radians are a way to measure angles, similar to degrees, but based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians use the circle's radius as the unit of measurement.
Imagine wrapping the radius of the circle around its circumference; the angle formed is 1 radian.
A full circle is equivalent to 2π radians, because the circumference of a circle is 2π times the radius. Understanding radians is crucial, as many mathematical and physics calculations use this measurement. Here are some key things to remember about radians:

• 1 radian is approximately 57.3 degrees.
• A half circle or semicircle is π radians.
• Quarter circles are π/2 radians.

Using radians can simplify problems involving arc lengths, sectors, and rotations, making it a preferred unit of angle measurement in advanced math.

Angle Range

When we are tasked with finding an angle within a specified range, understanding the concept of an angle range becomes essential. In this particular exercise, the range is from -2π to 0 radians.
This means that any coterminal angle we’re looking to find must fall between these two boundaries.To correctly place an angle within a given range, you often need to adjust the angle by adding or subtracting a full circle (2π radians). For instance, if we start with the angle \(rac{5 ext{π}}{6}\), we know it is outside the -2π to 0 range because it's positive.
Subtracting 2π will help us 'wrap' or adjust it back into the specified range. Here is how to work with angle ranges effectively:

• Identify the target range where your angle should lie.
• Use full rotations to adjust your angle by adding or subtracting.
• Verify that the newly adjusted angle correctly fits within the range.

Full Rotations

Full rotations in the context of angles involve adding or subtracting multiples of a complete circle. A full rotation is quantified by 2π radians. This concept helps us identify angles that look the same on a circle, known as coterminal angles.
When dealing with full rotations, understanding that angles separated by these rotations share the same terminal side is key. For example, \(rac{5 ext{π}}{6}\) and -\(rac{7 ext{π}}{6}\) are coterminal, even though they are not numerically identical. Subtracting 2π from \(rac{5 ext{π}}{6}\) gives us an angle that has completed a full circle (or rotation).This is useful in applications such as trigonometry and physics, where rotation angles go beyond 360 degrees or 2π radians. Here's how to approach full rotations:

• Identify if the given angle can be adjusted within the desired range.
• Add or subtract 2π to attempt to "wrap" the angle into the desired interval.
• Ensure the new angle maintains the coterminality, meaning its terminal position on the circle remains the same.