Problem 7 - 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. \(405^{\circ}\)

Step-by-Step Solution

Answer

The reference angle for a 405° angle is 45°.

Step 1: Understand the Standard Position

An angle is in standard position if its vertex is at the origin of the coordinate plane, and its initial side lies along the positive x-axis. The terminal side represents the angle's rotation.

Step 2: Determine the Coterminal Angle

For angles larger than 360°, find a coterminal angle between 0° and 360°. Do this by subtracting 360° from the original angle until the result is between 0° and 360°: \[ 405° - 360° = 45° \]. Thus, a 405° angle is coterminal with a 45° angle.

Step 3: Draw the Angle in Standard Position

Draw the 45° angle (coterminal angle) in standard position. Start from the positive x-axis and rotate 45° counterclockwise.

Step 4: Understand the Concept of Reference Angle

A reference angle is the acute angle formed by the terminal side of the given angle and the x-axis.

Step 5: Draw the Reference Angle

Since 45° is already an acute angle and is the measure between the terminal side and the x-axis, the reference angle is itself. Draw the same position and emphasize what makes it the reference angle.

Step 6: Measure the Reference Angle

The reference angle of a 45° angle, which in this case is the angle itself, is \[ 45° \].

Key Concepts

Reference Angle

A reference angle is a key concept when studying angles in trigonometry. It is essentially the smallest angle that you can form between the terminal side of an angle and the x-axis.
The reference angle will always be a positive acute angle, which means it's less than 90°. This property of being acute helps simplify calculations and analyses of angles on the coordinate plane.

To find a reference angle:

• Identify the terminal side of the angle you're analyzing.
• Measure the angle to the x-axis.

For an angle like 405°, which simplifies to a coterminal angle of 45°, the reference angle is simply the angle itself, 45°. Since this is already less than 90°, it's considered acute and serves as its own reference angle.

Standard Position

Angles are often positioned in what is known as standard position. This provides a way to consistently and conveniently draw and interpret angles on the coordinate plane.
In standard position, the angle's vertex is placed exactly at the origin of the coordinate plane. Its initial side aligns perfectly with the positive x-axis.

Here's how to place an angle in standard position:

• The vertex of the angle sits at the origin \(0,0\).
• The initial side lies along the positive x-axis.
• The terminal side will represent where the angle ends after rotating counterclockwise (for positive angles) from the initial side.

For example, a 45° angle in standard position would start from the positive x-axis and extend 45° counterclockwise. This method of placing angles aids in visualizing trigonometric concepts easily.

Acute Angle

An acute angle is a special type of angle that measures less than 90°. It stands out for several reasons in geometry and trigonometry.
Acute angles are essential as they represent the smallest angles in geometry. They play a crucial role in identifying reference angles, as reference angles are always acute.

Key characteristics of acute angles include:

• Always less than 90°.
• Appear sharp and narrow.
• Used frequently in trigonometry because of their simplicity.

If you have a 45° angle, it is already an acute angle since it satisfies the condition of being less than 90°. Such angles are easy to work with and serve as reference angles without further adjustments.

Coordinate Plane

When dealing with angles, a coordinate plane becomes a fundamental tool. It provides a graphical way to see angles and understand their properties.
The coordinate plane consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where they intersect is known as the origin.

Using the coordinate plane for angles involves:

• Placing the angle's vertex at the origin.
• Aligning the initial side of the angle with the positive x-axis.
• Allowing the terminal side to express the angle based on its rotation.

This setup helps in visualizing how an angle, such as 405°, reduces to a simple 45° when considered in its coterminal form, making analysis and computations more intuitive.