Problem 111 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

Thus the radian measure of two full revolutions clockwise is \(-4\pi\) radians.

Step 1: Understand Radian Measure

Begin by acknowledging that a full revolution around a circle equates to \(2\pi\) radians. This means the measure of angle when one full revolution is \(2\pi\) radians.

Step 2: Understand Direction of Rotation

Here, the rotations are clockwise which in mathematical terms is considered as negative rotation. Therefore, an angle generated by a full revolution clockwise will be \(-2\pi\) radians.

Step 3: Calculate Total Radian Measure

Now, for two full revolutions, the total radian measure of the angle will simply be two times \(-2\pi\) radians, which equals to \(-4\pi\) radians.

Key Concepts

Angle in Standard Position

An angle in standard position is a fundamental concept in trigonometry. To define an angle in standard position, you'll start with three key components:

• The angle's vertex is positioned at the origin of a coordinate plane.
• Its initial side lies along the positive x-axis.
• The terminal side, where the angle opens out to, can move into any quadrant depending on the size of the angle.

When calculating angles, especially when using radian measure, setting them in standard position allows for consistency. It's essential because it provides a reference point, making it easier to understand and calculate rotations or measures related to the angle. Regardless of the direction or magnitude of rotation, an angle's position starts the same when it's in standard position.

Clockwise Rotation

In the study of rotations, direction is key. Clockwise rotation moves in the direction of the hands of a clock. In mathematics, this direction is typically considered negative. This "negative" notation helps distinguish clockwise from the standard positive direction of counterclockwise rotation.

When dealing with problems involving the radian measure, as in the original exercise, it's important to note this negative value. Moving clockwise through one full revolution results in an angle measure of \(-2\pi\) radians. Identifying the direction of rotation makes calculations more accurate and ensures we're interpreting the angle correctly within the context of its position and movement.

Full Revolution

A full revolution refers to one complete 360-degree turn around a circle. In the world of radian measure, this is equivalent to \(2\pi\) radians. This value is crucial because it represents the entirety of a circle's circumference as applied to angle measures.

Understanding full revolutions in terms of radians is key when calculating far or repeated rotations. For every additional full rotation, the angle measure increases by \(2\pi\) radians clockwise or counterclockwise. In the context of the original exercise, two full clockwise revolutions total to \(-4\pi\) radians because it is twice around the circle moving in the negative (clockwise) direction, each full rotation adding \(-2\pi\). Keeping track of these rotations allows mathematicians to measure and understand angles larger than \(360\) degrees efficiently.