Problem 5 - 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. \(320^{\circ}\)

Step-by-Step Solution

Answer

The reference angle for 320° is 40°.

Step 1: Draw the Angle in Standard Position

To draw the angle in standard position, start by marking the initial side on the positive x-axis. Then measure and rotate counterclockwise 320° from this position. This rotation will end the terminal side in the fourth quadrant since 320° is more than 270° but less than 360°.

Step 2: Identify the Reference Angle

The reference angle is the positive acute angle between the terminal side of our angle and the x-axis. For angles in the fourth quadrant, the reference angle can be found by subtracting the angle from 360°. Thus, the reference angle is given by 360° - 320° = 40°.

Step 3: Draw the Reference Angle as an Acute Angle

In the fourth quadrant, draw the reference angle measuring 40° from the terminal side of the original angle to the x-axis. This acute angle should be less than 90° and should clearly show the difference from 360°.

Step 4: Draw the Reference Angle in Standard Position

To draw the reference angle in standard position, draw a 40° angle starting from the positive x-axis and moving counterclockwise. This involves placing its terminal side in the first quadrant, where all angles below 90° reside.

Step 5: Measure the Reference Angle

The measure of the reference angle is obtained from our calculation in Step 2. Clearly, the reference angle for 320° in standard position is 40°.

Key Concepts

Standard Position

When we talk about angles in standard position, we mean an angle that is situated in a particular way on a coordinate plane.
An angle is considered to be in standard position if:

• The vertex is at the origin, where the x-axis and y-axis intersect.
• The initial side, or one of the legs of the angle, lies on the positive x-axis.
• The angle itself is measured from the initial side.

Typically, angles are measured in a counterclockwise direction from the initial side. This direction is considered positive. For example, when you have a 320° angle, like in our exercise, it means you start from the positive x-axis and measure a 320° counterclockwise rotation, placing the terminal side in the fourth quadrant.
Remember, always follow this setup for depicting angles in standard position, which simplifies your understanding and calculations.

Acute Angles

Acute angles are a type of angle that are always less than 90°.
They are small, distinct, and often easier to work with compared to larger angles.

• The fundamental property of an acute angle is that it falls between 0° and 90°, which means it "opens up" only a little.
• In the context of reference angles, an acute angle is formed between the terminal side of the given angle and the x-axis.

For a 320° angle, the reference angle formed with respect to the x-axis is precisely 40°, which falls well within the acute angle category. This 40° angle offers a useful way to visualize and solve trigonometric problems since many trigonometric functions use reference angles to simplify calculations.

Angle Measurement

Understanding angle measurement is key to resolving trigonometric problems. Angles can be measured in degrees or radians, though here we will focus on degrees, as that's the method used in most basic geometry and trigonometry problems.
A full circular rotation is 360°, which serves as a useful reference point for understanding larger angles.

• Angles larger than 90° can often be broken down into smaller, more manageable parts such as reference angles.
• For instance, an angle like 320° is understood by identifying the smaller reference angle (40°) that helps relate it back to more familiar acute angles.

By breaking down complex angles using their reference angles, students can simplify their calculations and gain a clearer understanding of how angles relate to circular paths and trigonometric functions. This breakdown not only aids in drawing but also enhances comprehension and problem-solving skills.