Directed rotation and coterminal angles

Solution

Begin by identifying the mathematical object and the information that fixes it. Combine successive rotations algebraically, then reduce the net angle. For all coterminal angles, write $theta+360k$ degrees or $theta+2kpi$ radians, where $k$ is any integer. The relevant conditions are not optional bookkeeping: State the interval requested. The smallest positive coterminal angle lies in $(0,360]$ degrees or $(0,2pi]$ radians, while a principal standard-position angle is often chosen in $[0,360)$ or $[0,2pi)$. Following that structure gives Net rotation is $585$ degrees, coterminal with $225$ degrees.

Why this works

To reduce an angle, add or subtract complete revolutions until the value lies in the desired interval. The operation changes the accumulated rotation but preserves the terminal side because every full revolution returns the ray to its prior direction. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Worked example 1

A radar arm begins on the positive x-axis, rotates $765$ degrees counterclockwise, then $180$ degrees clockwise. Where is its terminal side, and what is the smallest positive coterminal angle?

Solution

Begin by identifying the mathematical object and the information that fixes it. Combine successive rotations algebraically, then reduce the net angle. For all coterminal angles, write $theta+360k$ degrees or $theta+2kpi$ radians, where $k$ is any integer. The relevant conditions are not optional bookkeeping: State the interval requested. The smallest positive coterminal angle lies in $(0,360]$ degrees or $(0,2pi]$ radians, while a principal standard-position angle is often chosen in $[0,360)$ or $[0,2pi)$. Following that structure gives Net rotation is $585$ degrees, coterminal with $225$ degrees.

Why this works

To reduce an angle, add or subtract complete revolutions until the value lies in the desired interval. The operation changes the accumulated rotation but preserves the terminal side because every full revolution returns the ray to its prior direction. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Problem

Convert $\frac{11}{6}$ revolutions to degrees and locate the terminal side.

Worked development

Combine successive rotations algebraically, then reduce the net angle. For all coterminal angles, write $theta+360k$ degrees or $theta+2kpi$ radians, where $k$ is any integer. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. The same terminal side can be reached after any whole number of revolutions. In degrees, one revolution adds $360$ degrees; in radians, it adds $2pi$. This means an angle is not determined by its terminal side alone: $45$ degrees, $405$ degrees, and $-315$ degrees describe different rotations with the same final direction. Then apply the conditions explicitly: State the interval requested. The smallest positive coterminal angle lies in $(0,360]$ degrees or $(0,2pi]$ radians, while a principal standard-position angle is often chosen in $[0,360)$ or $[0,2pi)$. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Directed rotations appear in navigation, rotating machinery, phase, circular motion, and complex-number arguments.

Problem

Find all angles coterminal with $-140$ degrees.

Worked development

Combine successive rotations algebraically, then reduce the net angle. For all coterminal angles, write $theta+360k$ degrees or $theta+2kpi$ radians, where $k$ is any integer. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. The same terminal side can be reached after any whole number of revolutions. In degrees, one revolution adds $360$ degrees; in radians, it adds $2pi$. This means an angle is not determined by its terminal side alone: $45$ degrees, $405$ degrees, and $-315$ degrees describe different rotations with the same final direction. Then apply the conditions explicitly: State the interval requested. The smallest positive coterminal angle lies in $(0,360]$ degrees or $(0,2pi]$ radians, while a principal standard-position angle is often chosen in $[0,360)$ or $[0,2pi)$. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

Directed rotations appear in navigation, rotating machinery, phase, circular motion, and complex-number arguments.

Worked example 4: quick check

Give one positive and one negative angle coterminal with $310$ degrees.

Solution

Begin by identifying the mathematical object and the information that fixes it. Combine successive rotations algebraically, then reduce the net angle. For all coterminal angles, write $theta+360k$ degrees or $theta+2kpi$ radians, where $k$ is any integer. The relevant conditions are not optional bookkeeping: State the interval requested. The smallest positive coterminal angle lies in $(0,360]$ degrees or $(0,2pi]$ radians, while a principal standard-position angle is often chosen in $[0,360)$ or $[0,2pi)$. Following that structure gives $670$ degrees and $-50$ degrees.

Why this works

To reduce an angle, add or subtract complete revolutions until the value lies in the desired interval. The operation changes the accumulated rotation but preserves the terminal side because every full revolution returns the ray to its prior direction. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Connection forward

The next lesson turns one revolution into a numerical measuring system using degrees.