Problem 47 - Precalculus: A Concise Course

Question

Sketch each angle in standard position. (a) \(-30^{\circ}\) (b) \(-135^{\circ}\)

Step-by-Step Solution

Answer

The angles are sketched in standard positions. Angle \(-30^{\circ}\) is located in the 4th quadrant, and angle \(-135^{\circ}\) is located in the 2nd quadrant.

Step 1: Understanding What Standard Position of an Angle Means

Firstly, it's necessary to understand that an angle is said to be in standard position if its vertex is at the origin and one ray is on the positive x-axis. If the angle is positive, rotation from the positive x-axis is counterclockwise. If the angle is negative, rotation from the positive x-axis is clockwise.

Step 2: Sketching \(-30^{\circ}\)

The angle is negative, so the rotation will be clockwise from the positive x-axis. To sketch, starting from the positive x-axis, move clockwise with an angle measure of 30 degrees. It will be located in the 4th quadrant on any standard Cartesian coordinate system as it is a negative angle.

Step 3: Sketching \(-135^{\circ}\)

Again, because the angle is negative, the rotation will be clockwise from the positive x-axis. Starting from the positive x-axis, move clockwise with an angle measure of 135 degrees. This will lead into the 2nd quadrant on any standard Cartesian coordinate system.

Key Concepts

Standard Position of an Angle

When we talk about the standard position of an angle, it means that the angle has its vertex at the origin of a coordinate system, and one of its sides, known as the initial side, is aligned with the positive x-axis. The other side, called the terminal side, is where the angle 'opens' up to. To visualize it:

• Imagine standing at the center of a clock facing 12 o'clock—this is like the initial side.
• When you pivot or rotate, your outstretched arm points in the direction of the terminal side.

If we follow this guideline, any positive or negative rotation can be easily sketched onto a Cartesian coordinate plane, facilitating a student's comprehension of the geometric concept.

Negative Angle Rotation

Understanding negative angle rotation can initially seem counterintuitive, but it's like rewinding a movie. Here's how to picture it:

• Positive angles rotate in a counterclockwise direction, which is the 'forward' movement.
• Negative angles rotate in the opposite direction, which is clockwise, akin to 'rewinding.'

Envisioning the counterclockwise movement as adding time and clockwise as taking time away can help solidify the concept. Applying this to the Cartesian coordinate system, a -30° angle, for example, will be sketched by turning 30° to the right of the 12 o'clock direction, landing in the 4th quadrant. Such visual aids can significantly enhance a student's ability to grasp and apply the idea of negative angles.

Cartesian Coordinate System

The Cartesian coordinate system is a fundamental concept that forms the basis of high school and college mathematics and physics. Named after René Descartes, here's what makes it special:

• It's made up of two perpendicular lines, normally labelled the x (horizontal) and y (vertical) axes.
• Where they meet, at the point (0, 0), is the origin.
• Each point on the plane is determined by an (x,y) pair, showing how far along and up you go from the origin.

For sketches of angles in standard position, this system allows precise location of the terminal side. If the angle is negative, as in the examples provided, it rotates clockwise, and the terminal side will reside in the quadrants typically encountered when moving to the left and down from the positive x-axis. When students familiarize themselves with this grid system, plotting angles, whether positive or negative, becomes straightforward.