Problem 15 - Algebra 2 and Trigonometry

Question

In \(8-17,\) for each angle with the given degree measure, find the measure of the reference angle. \(290^{\circ}\)

Step-by-Step Solution

Answer

The reference angle is \(70^{\circ}\).

Step 1: Determine Quadrant

The given angle is \(290^{\circ}\). Since a full rotation is \(360^{\circ}\), angles between \(270^{\circ}\) and \(360^{\circ}\) are in the fourth quadrant. Thus, \(290^{\circ}\) is in the fourth quadrant.

Step 2: Calculate Reference Angle

The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. For angles in the fourth quadrant, the reference angle \(\theta_{ref}\) can be found using \(360^{\circ} - \theta\). Thus, \(\theta_{ref} = 360^{\circ} - 290^{\circ} = 70^{\circ}\).

Key Concepts

Reference Angle

In trigonometry, the concept of a reference angle is pivotal for simplifying calculations and understanding the position of angles on the coordinate plane. A reference angle, denoted as \(\theta_{ref}\), is the smallest acute angle that the terminal side of a given angle makes with the x-axis. This angle always has a value between \(0^{\circ}\) and \(90^{\circ}\), which makes computations consistent and easier to handle. To find the reference angle, identify the quadrant in which the given angle lies. For each of these quadrants:

• **First Quadrant:** The reference angle is the angle itself.
• **Second Quadrant:** Subtract the angle from \(180^{\circ}\).
• **Third Quadrant:** Subtract \(180^{\circ}\) from the angle.
• **Fourth Quadrant:** Subtract the angle from \(360^{\circ}\).

Reference angles are essential in trigonometry for evaluating trigonometric functions and making the calculations less complex.

Fourth Quadrant

The fourth quadrant of the Cartesian plane is an essential area when determining the behavior of angles and their respective trigonometric functions. This quadrant is located in the bottom right section and includes all angles between \(270^{\circ}\) and \(360^{\circ}\). Here are some properties and attributes of the fourth quadrant:

• **Angles**: Angles in this quadrant decrease as they approach \(360^{\circ}\).
• **Sign**: In the fourth quadrant, the sine is negative, while the cosine and tangent functions are positive and negative, respectively.

For angles in the fourth quadrant, such as our example of \(290^{\circ}\), one can utilize the formula \(\theta_{ref} = 360^{\circ} - \theta\) to determine the reference angle, simplifying the analysis of trigonometric functions and angle measures.

Acute Angle

An acute angle is defined as any angle less than \(90^{\circ}\). In trigonometry, acute angles are fundamentally straightforward as they comfortably fit within the first quadrant, where both x and y values are positive. This simplicity is particularly helpful because the trigonometric functions (sine, cosine, and tangent) all retain positive values, and calculations are generally straightforward without the need for adjustments for negative values. Every reference angle is an acute angle, making it easier to apply trigonometric principles consistently across different quadrants.By converting the angle from larger measures to an acute angle, it allows a deeper understanding and an immediate visual and functional grasp of the angle’s behavior with respect to the x-axis.

Degree Measure

The degree measure in trigonometry refers to how angles are commonly represented. A complete revolution forms a circle of \(360^{\circ}\). Understanding the degree measure provides insight into how angles progress on the coordinate plane. Each quadrant contains angles spanning \(90^{\circ}\). Specifically, angles are located as follows:

• **First Quadrant**: \(0^{\circ}\) to \(90^{\circ}\)
• **Second Quadrant**: \(90^{\circ}\) to \(180^{\circ}\)
• **Third Quadrant**: \(180^{\circ}\) to \(270^{\circ}\)
• **Fourth Quadrant**: \(270^{\circ}\) to \(360^{\circ}\)

For example, an angle measuring \(290^{\circ}\) falls within the fourth quadrant. The degree measure helps identify not only the location of the angle but also its standard position, starting from the positive x-axis and moving counterclockwise. This measurement is central for determining reference angles and for working with trigonometric identities and functions.