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Chapter 7: Problem 16

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

Short Answer

Expert verified
540掳 in standard position is equivalent to 180掳.

Step by step solution

01

Convert Degree to Standard Angle

First, convert the angle from outside the range of a standard position to within the range of 0掳 to 360掳. Subtract multiples of 360掳 from 540掳 until the result is within this range: \ \[ 540^\text{掳} - 360^\text{掳} = 180^\text{掳} \]
02

Locate the Angle on the Coordinate Plane

The resulting angle of 180掳 needs to be drawn in standard position. Start from the positive x-axis and rotate counterclockwise by 180掳.
03

Draw the Angle

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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

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