Problem 109 - Precalculus : Building Concepts and Connections

Question

Find the radian measure of an angle in standard position that is generated by the specified rotation. One-third of a full revolution counterclockwise

Step-by-Step Solution

Answer

\(\frac{2\pi}{3}\) radians

Step 1: Understanding a Full Revolution

A full revolution in the Cartesian coordinate system or a circle represents a total radian measure of \(2\pi\). This equivalence is important in converting between radians and revolution fractions.

Step 2: Calculating the Radian Measure

Now we want to find the radian measure for one-third of a full revolution. Since a full revolution is \(2\pi\) radians, one-third of a full revolution will be \(\frac{2\pi}{3}\) radians.

Key Concepts

Full Revolution

In the context of circles and angles, a full revolution is when an angle covers a complete circle, starting from a fixed point and moving all the way around back to that same point. This is equivalent to 360 degrees in terms of angle measures. However, when dealing with radians, a full revolution measures exactly \(2\pi\) radians.

Radians are often preferable in many mathematical contexts, particularly calculus and trigonometry, because they provide a direct link between the radius of a circle and the arc length.

• A full circle: \(360^\circ\) or \(2\pi\) radians
• Half circle: \(180^\circ\) or \(\pi\) radians
• Quarter circle: \(90^\circ\) or \(\frac{\pi}{2}\) radians

Understanding a full revolution as \(2\pi\) radians is important in solving problems involving angular movement, paving the way to comprehend fractional parts of revolutions, like one-third or one-quarter revolutions.

Standard Position

An angle in standard position refers to an angle placed on the coordinate plane whereby its vertex is at the origin (0,0) of the Cartesian coordinate system, and its initial side lies along the positive x-axis. The angle is then measured in terms of a rotation from this initial side.

The direction of angle measurement is crucial and determines the sign of the angle:

• Counterclockwise rotation: This is considered as positive direction.
• Clockwise rotation: This is considered as negative direction.

Standard position allows for a consistent and standard way to describe angles when working with trigonometric functions, graphing, and solving equations related to angles and circles. Visualizing angles in standard position helps in understanding the concepts of angular displacements and rotational equivalence, which are foundational in trigonometry.

Fraction of Revolution

A fraction of revolution refers to a portion or a segment of a full circle or rotation. When dealing with angles, especially in radians, it involves calculating the radian measure corresponding to a specified fraction of a full turn.

Given that a full revolution is \(2\pi\) radians, calculating fractions like one-third requires multiplying \(2\pi\) by the given fraction. For instance:

• One-third of a full revolution: Multiply \(2\pi\) by \(\frac{1}{3}\), resulting in \(\frac{2\pi}{3}\) radians.
• One-fourth of a full revolution: Multiply \(2\pi\) by \(\frac{1}{4}\), resulting in \(\frac{\pi}{2}\) radians.

This concept is pivotal because it connects linear thinking (fractions of a whole) with angular displacement, a key in geometry, engineering, and physics for describing periodic motion or rotations.