Problem 108 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

Half of a full revolution counterclockwise is equal to \( \pi \) radians.

Step 1: Recognize the angle of rotation

First, we need to acknowledge that half a revolution, regardless of the direction, is equivalent to 180 degrees.

Step 2: Convert degrees to radians

Next, use the formula to convert degrees to radians. The formula is \( radians = degrees*\(\frac{\pi}{180}\) \). Apply this to 180 degrees, \( radians = 180*\(\frac{\pi}{180}\) \). Upon simplification, this equals \( \pi \) radians.

Step 3: The final answer

Therefore, half a full revolution counterclockwise is equal to \( \pi \) radians.

Key Concepts

Angle of Rotation

An angle of rotation refers to the amount of turn between two rays with a common end point, called the vertex. This concept is vital in mathematics and physics as it describes the orientation of one line to another. In particular, half of a full revolution means the rays have rotated through a half-circle from their starting position. To visualize this, imagine the hand of a clock starting at the 12 o'clock position and sweeping all the way to the 6 o'clock position - it has gone through half of a full turn, or 180 degrees.

An angle's rotation can be described in degrees, a unit well-known in daily life and geometry, or radians, commonly used in calculus and higher mathematics for its natural connection to the properties of circles. It is crucial for students to understand how to interpret this movement and measure it both in degrees and radians to apply it in various mathematical contexts.

Degrees to Radians Conversion

Converting from degrees to radians is an essential skill in mathematics. To do this, we use a specific conversion factor. Since a full circle is 360 degrees, and equivalently, \( 2\text{π} \) radians, we can easily convert by using the formula: \

\( radians = degrees * \frac{\text{π}}{180} \).

For example, to convert 180 degrees to radians, one would calculate \
\( 180 * \frac{\text{π}}{180} \),
which simplifies to \text{π}\ radians. It's essential to have this conversion process down as it can then be applied to any angle measure.

Revolution in Radians

A revolution refers to a full rotation or turn around a circle which is \
360 degrees or \( 2\text{π} \) radians. When we discuss fractions of a revolution, such as half a full revolution, we are essentially dividing the full radian measure of the circle. Since one revolution is \( 2\text{π} \) radians,