Problem 14 - Algebra 2 and Trigonometry

Question

In \(8-17,\) name the quadrant in which an angle of each given measure lies. $$ -280^{\circ} $$

Step-by-Step Solution

Answer

The angle \(-280^{\circ}\) lies in Quadrant I.

Step 1: Understanding the Problem

We are given an angle measure of \(-280^{\circ}\) and need to determine in which quadrant this angle lies on the Cartesian plane. Quadrants are numbered I through IV, starting from the positive x-axis and moving counterclockwise.

Step 2: Convert the Angle to Positive

To determine the quadrant, convert the angle to a positive equivalent by adding multiples of \(360^{\circ}\) until you have a positive angle. For \(-280^{\circ}\), add \(360^{\circ}\): \(-280^{\circ} + 360^{\circ} = 80^{\circ}\).

Step 3: Determine the Quadrant

An angle of \(80^{\circ}\) lies in the first quadrant, as angles from \(0^{\circ}\) to \(90^{\circ}\) fall in Quadrant I.

Key Concepts

Cartesian Plane

The Cartesian plane is a two-dimensional plane formed by two perpendicular lines which intersect at a point called the origin. These two lines are known as the x-axis and y-axis. Here's how it all fits together:

• The x-axis runs horizontally, and the y-axis runs vertically.
• The point where these two axes intersect is known as the origin, denoted as \( (0,0) \).
• The plane is divided into four different regions called quadrants.
• Quadrants are numbered counterclockwise starting from the positive x-axis.

Angles are plotted on the Cartesian plane, and each quadrant corresponds to a specific range of angle measurements:

• Quadrant I: 0° to 90°
• Quadrant II: 90° to 180°
• Quadrant III: 180° to 270°
• Quadrant IV: 270° to 360°

These classifications help us understand what direction an angle points, or where points lie in relation to the origin on the plane.

Angle Conversion

Angle conversion is an essential process in trigonometry. It allows us to deal with angles more flexibly, particularly converting negative angles to positive ones. This is crucial when determining which quadrant an angle falls into. Here's how you can convert angles:- **Understanding Negative Angles**: Negative angles are measured clockwise from the positive x-axis, unlike positive angles which are measured counterclockwise. They can lead to confusion when landed the coordinate plane.- **The 360° Trick**: To convert a negative angle to its positive equivalent, simply add 360° (or a multiple of 360°) until the angle is positive. For example, with an angle of \( -280^\circ \), adding 360° results in 80°, which is positive.This conversion is useful because it simplifies calculations and makes it easy to determine which quadrant an angle is in. By converting to an equivalent positive angle, we maintain the same direction on the plane, ensuring our trigonometric calculations remain consistent and accurate.

Positive Angle

Understanding positive angles is critical in determining the position on the Cartesian plane. Positive angles are measured counterclockwise from the positive x-axis. They are easier to manage because they fit directly with how quadrants are numbered. Here's what makes an angle "positive":

• Measured counterclockwise from the positive x-axis.
• Range from 0° to 360°.
• Directly indicates which quadrant it belongs to based on its measurement.

In practical problems, a positive angle makes it easier to relate to the familiar settings of the quadrants. Furthermore, when dealing with trigonometric functions, knowing the positive representation of an angle simplifies graphing and calculations.So when you're asked for the quadrant of an angle like \( -280^\circ \), converting to a positive angle \( (80^\circ) \) immediately identifies Quadrant I as its location. This handling of angles ensures a strong fundamental comprehension of geometric and trigonometric concepts.