Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-7,\) for each angle with the given degree measure: a. Draw the angle in standard position b. Draw its reference angle as an acute angle formed by the terminal side of the angle and the \(x\) -axis. c. Draw the reference angle in standard position. d. Give the measure of the reference angle. \(120^{\circ}\)

Step-by-Step Solution

Answer

The reference angle for \(120^{\circ}\) is \(60^{\circ}\).

Step 1: Identify the Quadrant

Since the angle given is \(120^{\circ}\), which is more than \(90^{\circ}\) and less than \(180^{\circ}\), it is located in the second quadrant.

Step 2: Draw Angle in Standard Position

In standard position, the initial side of the angle is along the positive \(x\)-axis. Starting from there, rotate counterclockwise to create the \(120^{\circ}\) angle in the second quadrant.

Step 3: Find the Reference Angle

The reference angle is the acute angle formed between the terminal side of the angle and the \(x\)-axis. Since the angle is in the second quadrant, the reference angle is: \(180^{\circ} - 120^{\circ} = 60^{\circ}\).

Step 4: Draw the Reference Angle

To represent the reference angle in standard position, draw a line from the positive \(x\)-axis, rotating \(60^{\circ}\) counterclockwise. This imaginary angle is useful for calculations.

Step 5: Provide Measure of Reference Angle

The measure of the reference angle is \(60^{\circ}\), which is the acute angle formed with the \(x\)-axis.

Key Concepts

Standard Position

Angles in mathematics are often drawn in what is known as the 'standard position'. This term defines a specific way to represent angles on the coordinate plane. Here’s how it works:

• The vertex of the angle is placed at the origin of the coordinate system, where the x-axis and y-axis meet.
• The initial side of the angle always lies along the positive x-axis. This is where the angle starts.
• From the initial side, the angle is measured by rotating counterclockwise to a new position.

This setup helps us consistently define the position of angles and is particularly useful when identifying the quadrant an angle lies in. For instance, an angle of \(120^{\circ}\) in standard position starts from the positive x-axis and extends counterclockwise through the second quadrant.
This consistent method of representing angles makes calculations and predictions more straightforward.

Terminal Side

In geometry, when we discuss angles, the 'terminal side' is a critical concept to understand. The terminal side is simply the side of the angle that rotates from the initial side.

• After leaving the initial side along the positive x-axis, the angle swings to land on the terminal side.
• The position at which the terminal side lands determines the angle's measurement, quadrant, and reference angles.
• In the example of a \(120^{\circ}\) angle, the terminal side lands in the second quadrant.
  The terminal side is crucial because it directly interacts with the x-axis, making it possible to determine reference and acute angles.

Knowing where the terminal side is can help in identifying properties and other angles related to the original.

Second Quadrant

The coordinate plane is divided into four main sections, called quadrants, each representing different signs for x and y coordinates. The second quadrant is one of these sections.

• In the second quadrant, the x-values are negative while the y-values are positive.
• Angles between \(90^{\circ}\) and \(180^{\circ}\) fall into this quadrant when drawn in standard position.
• For a \(120^{\circ}\) angle, which is more than \(90^{\circ}\) but less than \(180^{\circ}\), the terminal side sits comfortably in the second quadrant.

Understanding which quadrant an angle falls into helps in determining the reference angles and the signs of trigonometric functions.

Acute Angle

An acute angle is one of the three primary types of angles in geometry, characterized by its measure.

• Acute angles are always less than \(90^{\circ}\).
• When working with standard position angles, the acute angle formed with the x-axis is often called the reference angle.
• For our \(120^{\circ}\) angle, the reference angle is \(60^{\circ}\), which is an acute angle because it's less than \(90^{\circ}\).

Understanding acute angles is key to resolving problems that involve angles in geometry and trigonometry.
They help simplify calculations when determining trigonometric function values for angles outside the first quadrant.