Problem 8 - Precalculus

Question

Sketch each angle in standard position. (a) \(450^{\circ}\) (b) \(-180^{\circ}\) (c) \(270^{\circ}\)

Step-by-Step Solution

Answer

Sketch 90°, 180°, and 270° angles starting from the positive x-axis.

Step 1: Determine the Reference Angle

For an angle, the reference angle can be found by finding the equivalent angle between 0° and 360°. If the given angle is greater than 360°, subtract 360° until it falls within the 0° to 360° range. If it is negative, keep adding 360° until it becomes positive.

Step 2: Sketch the Angle 450°

For 450°, subtract 360° to find the equivalent angle: \[450° - 360° = 90°\] Sketch a 90° angle in standard position on the coordinate plane, starting from the positive x-axis to the positive y-axis.

Step 3: Sketch the Angle -180°

For -180°, add 360° to find the equivalent angle: \[-180° + 360° = 180°\] Sketch a 180° angle in standard position, which is a straight line extending from the positive x-axis to the negative x-axis along the diameter of a circle.

Step 4: Sketch the Angle 270°

For 270°, the angle directly falls between 0° and 360°, hence no need for conversion. Sketch a 270° angle in standard position on the coordinate plane, starting from the positive x-axis to the negative y-axis.

Key Concepts

Reference Angle

A reference angle is a crucial part of understanding angles on a coordinate plane. It helps simplify the process of sketching and working with angles. The reference angle is always found between the original angle and the nearest x-axis.

• To determine the reference angle, you must bring the given angle into the standard interval of 0° to 360°.
• For angles greater than 360°, subtract 360° repeatedly until it falls within 0° and 360°.
• If the angle is negative, add 360° repeatedly until the angle is positive.

Reference angles are useful because they give you a smaller, more manageable angle without changing the trigonometric properties relative to the initial angle. This simplifies calculations and aids in visualizing the angle on a graph.

Standard Position

In trigonometry, sketching angles is often done in what is called the 'standard position'. This method provides a consistent reference point for measuring angles. An angle is in standard position when:

• The vertex of the angle is at the origin of the coordinate plane.
• The initial side lies along the positive x-axis.
• The angle is measured from the initial side, moving counterclockwise for positive angles and clockwise for negative angles.

This uniform way to draw angles allows for easy comparison and computation involving angles and shapes. It is foundational for understanding how angles interact in different mathematical contexts.

Coordinate Plane

The coordinate plane is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). It is crucial when working with angles in trigonometry.

• It supports visualizing angles by providing a framework where angles start from the origin and open outwards.
• Aids in understanding how angles relate to each of the four quadrants, with each quadrant representing a specific range of degree measures.
• Helps in determining the signs of trigonometric functions based on the position of an angle's terminal side.

Using the coordinate plane allows students to see the physical representation of angles, making abstract math concepts more concrete and understandable.

Positive and Negative Angles

Angles can be measured in either a positive or negative direction. This is a fundamental concept in trigonometry that affects how angles are sketched and understood.

• Positive angles are measured counterclockwise from the initial side on the positive x-axis.
• Negative angles are measured clockwise from the same initial side.
• Recognizing whether an angle is positive or negative is important when determining its direction and position on the coordinate plane.

Understanding the difference between positive and negative angles helps in accurately representing angles and solving problems where angle orientations matter.