Problem 46 - Algebra 2

Question

In which quadrant, or on which axis, does the terminal side of each angle lie? $$ 210^{\circ} $$

Step-by-Step Solution

Answer

The terminal side of an angle of 210 degrees lies in Quadrant III.

Step 1: Determine the division of degrees among quadrants

First, remember that the Cartesian coordinate system is divided into four quadrants, with the x and y axes dividing these quadrants. Quadrant I (where both x and y are positive) contains all angles from 0 to 90 degrees. Quadrant II (where x is negative and y is positive) holds angles from 90 to 180 degrees. Quadrant III (where both x and y are negative) includes angles from 180 to 270 degrees. And finally, Quadrant IV (where x is positive and y is negative) is for angles from 270 to 360 degrees.

Step 2: Compare the given angle with the degree range of each quadrant

The angle in question is 210 degrees. One must compare this with the degree range defined for each quadrant. It is seen that 210 degrees belongs to the range 180 to 270 degrees.

Step 3: Identify the quadrant

Since the 210 degrees falls in the range of 180 to 270 degrees, it therefore lies in Quadrant III where both x and y are negative.

Key Concepts

Coordinate System

When studying trigonometry, understanding the coordinate system is essential. The coordinate system is a grid that helps in locating points on a plane using two numbers. These numbers are called coordinates. It consists of two perpendicular lines, called the x-axis (horizontal) and y-axis (vertical), which intersect at a point called the origin.

The coordinate system is further divided into four quadrants. Each quadrant represents a unique combination of positive and negative values for coordinates. Knowing which quadrant a point lies in helps to determine the sign of trigonometric functions. This system is foundational in trigonometry for analyzing angles and their terminal sides.

Angles Measurement

Angles are measured in degrees and radians. In this context, we focus on degrees. An angle is formed by two rays (the initial side and the terminal side) originating from a common endpoint, called the vertex.

In trigonometry, angles can assume positive or negative values depending on their direction. A complete circle measures 360 degrees.

• Positive angles are measured counterclockwise from the initial side.
• Negative angles are measured clockwise.

Understanding how to measure angles is crucial for placing them correctly in the quadrants of the coordinate system.

Cartesian Coordinate System

The Cartesian coordinate system is a specific type of coordinate system that is of immense importance in trigonometry. Named after the mathematician René Descartes, it helps in visualizing algebraic equations.

Each point in this system is defined by a pair of numerical coordinates. The Cartesian plane is divided by the x and y axes, forming four quadrants.

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, y is negative.

The location of angles in these quadrants helps to determine various trigonometric function values.

Trigonometric Functions

Trigonometric functions are foundational in understanding the relationships between the angles and lengths in right triangles. These functions include sine, cosine, tangent, and others.

Each trigonometric function can have different signs depending on the quadrant in which its terminal side lies. For example:

• In Quadrant I, all trigonometric functions are positive.
• In Quadrant II, sine is positive. However, cosine and tangent become negative.
• In Quadrant III, tangent is positive, while sine and cosine are negative.
• In Quadrant IV, cosine is positive, while sine and tangent are negative.

Understanding the behavior of trigonometric functions in different quadrants is essential for solving trigonometric problems effectively.