Problem 112 - Precalculus : Building Concepts and Connections

Question

Find the radian measure of an angle in standard position that is generated by the specified rotation. Three full revolutions counterclockwise

Step-by-Step Solution

Answer

The radian measure of an angle in standard position generated by three full revolutions counterclockwise is \(6\pi\) radians.

Step 1: Determine the radian per revolution

In order to compute the total radians, we need to first know how many radians a full revolution generates. From knowledge of the properties of circles and radians, we know that one full revolution on a unit circle is equivalent to \(2\pi\) radians.

Step 2: Compute the radians for three revolutions

Since we know one revolution is equal to \(2\pi\) radians, then three revolutions would be \(3 \times 2\pi\) radians. We perform the multiplication.

Step 3: Deliver the solution

After performing the multiplication, we find that the counterclockwise rotation of 3 full revolutions is equivalent to \(6\pi\) radians.

Key Concepts

Standard Position

The concept of 'standard position' in trigonometry and geometry refers to a specific way of placing an angle on a coordinate plane. The vertex of the angle is positioned at the origin of the plane, and the initial side of the angle lies along the positive x-axis. This forms the baseline from which we measure the angle. It's very useful because it provides a consistent frame of reference for talking about angles and their measures. When we place angles in standard position, we can easily determine their trigonometric values and analyze their properties. This standardization also helps in comparing different angles to see how they relate to each other on the unit circle.

Unit Circle

The unit circle is a fundamental concept in trigonometry and often comes up in conversations about angles, especially those in standard position. The name "unit circle" comes from the circle's radius of one unit.
A circle centered around the origin in the coordinate plane helps visualize angle measures given in radians or degrees.

• Every point on the unit circle can be represented by coordinates \( (\cos(\theta), \sin(\theta)) \), where \ \theta \ is the angle in radians.
• These coordinates directly relate to the trigonometric functions sine and cosine.

Utilizing the unit circle, we can easily understand how a full rotation equates to a circumference of \(2\pi \) radians. That's why, for one complete spin around the circle—a full revolution—we use the value \(2\pi \). The unit circle thus serves as a bridge between angles, their measurement in radians, and trigonometric functions.

Full Revolution

A full revolution is an important metric in understanding circular motion and angle measurement, especially on the unit circle. When an object makes a complete circle, it has completed a full revolution.
For angles, one full revolution corresponds to an angle of \(360^\circ \) or \(2\pi \) radians.

• This concept is vital for converting between degrees and radians.
• In problems like the one described, calculating multiple revolutions involves simple multiplication.

For instance, three full revolutions would simply be \(3 \times 2\pi = 6\pi \) radians, as each revolution contributes \(2\pi \) radians to the total. Understanding this is key to mastering radians and solving related mathematical problems.

Counterclockwise Rotation

Counterclockwise rotation is the default direction of rotation in trigonometry when discussing angles in standard position. It is opposite to the direction in which clock hands move.
When measuring angles counterclockwise from the positive x-axis, we can determine if the angle is positive.

• Positive angles are measured counterclockwise.
• Negative angles, conversely, are measured clockwise.

In many trigonometric problems, such as the one giving rise to this discussion, a counterclockwise rotation often means adding positive multiples of \(2\pi\) to the angle, as seen with multiple revolutions. Understanding these rotations helps in determining the position and values of angles correctly on the unit circle.