Question

Draw each angle in standard position. \(540^{\circ}\)

Short answer

540° in standard position is equivalent to 180°.

Step-by-step solution

Convert Degree to Standard Angle

First, convert the angle from outside the range of a standard position to within the range of 0° to 360°. Subtract multiples of 360° from 540° until the result is within this range: \ \[ 540^\text{°} - 360^\text{°} = 180^\text{°} \]

Locate the Angle on the Coordinate Plane

The resulting angle of 180° needs to be drawn in standard position. Start from the positive x-axis and rotate counterclockwise by 180°.

Draw the Angle

Begin at the origin. Draw a line along the x-axis to the left (since 180° is a straight line). This line meets the negative side of the x-axis, completing the angle.

Key concepts

Angle Conversion

Angles greater than 360° or less than 0° need to be converted to fall within the standard 0° to 360° range.
This makes understanding and working with them easier.
To convert an angle, subtract or add multiples of 360° until the angle is within the desired range.
For example, with an angle of 540°, you subtract 360° (one full rotation) once to get 180°: \[ 540^\text{°} - 360^\text{°} = 180^\text{°} \]
This calculation brings the angle into the standard range.

Coordinate Plane

The coordinate plane is a key tool for visualizing angles.
It has two perpendicular lines known as axes: the x-axis (horizontal) and the y-axis (vertical).
The point where these axes intersect is called the origin.
Angles in standard position start at the origin and extend counterclockwise from the positive x-axis.
This makes it easy to accurately locate and draw angles.

Counterclockwise Rotation

When drawing angles in standard position, we typically use counterclockwise rotation.
This is the standard direction for measuring positive angles.
Starting from the positive x-axis, you rotate counterclockwise to reach the desired angle.
For 180°, this means rotating half-way around the coordinate plane, so the angle line ends on the negative x-axis.
This consistent method helps in accurately drawing and understanding angles on the coordinate plane.