Problem 11 - Algebra 2 and Trigonometry

Question

In \(8-17,\) for each angle with the given degree measure, find the measure of the reference angle. \(250^{\circ}\)

Step-by-Step Solution

Answer

The reference angle is 70°.

Step 1: Understand the Concept of Reference Angle

The reference angle is the acute angle formed by the terminal side of the given angle and the horizontal axis. It is always between 0° and 90°.

Step 2: Determine the Quadrant of the Angle

The angle given is 250°, which is greater than 180° but less than 270°. This means it's located in the third quadrant.

Step 3: Find the Nearest Horizontal Axis

The nearest horizontal axis in the third quadrant is 180°. To find reference angles in this quadrant, we calculate how far the angle extends beyond 180°.

Step 4: Calculate the Reference Angle

To find the reference angle for an angle in the third quadrant, subtract 180° from the given angle. The calculation is: 250° - 180° = 70°.

Key Concepts

Angle Measurement

Understanding angle measurement is essential in grasping how reference angles work. When we discuss angles in trigonometry, we often measure them in degrees. One full rotation around a circle equals 360 degrees. Any angle is essentially a measure of rotation from the initial side on the positive x-axis to the terminal side of the angle.

In trigonometry, angles can be positive or negative. A positive angle is measured counterclockwise from the initial side, while a negative angle is measured clockwise.

• Positive angles: Rotate counterclockwise
• Negative angles: Rotate clockwise

Understanding these basics helps in determining how far an angle reaches as it progresses around a circle, and is essential for calculating reference angles.

Quadrants in Trigonometry

The Cartesian plane is divided into four quadrants, each with its unique properties. Knowing which quadrant an angle is located in is crucial for determining its reference angle.

Here's a quick guide to understanding these quadrants:

• First Quadrant: Both x and y are positive. Angles between 0° and 90° fall here.
• Second Quadrant: x is negative, y is positive. Angles between 90° and 180° fall here.
• Third Quadrant: x and y are both negative. Angles between 180° and 270° exist here. For instance, a 250° angle is in this quadrant.
• Fourth Quadrant: x is positive, y is negative. Angles between 270° and 360° are found here.

Recognizing the quadrant allows you to find the reference angle by examining how far the original angle extends beyond the nearest horizontal axis.

Trigonometric Concepts

Trigonometry revolves around the study of angles and their relationships within a triangle, especially right triangles. Reference angles are vital in simplifying trigonometric calculations because they reduce larger angles to acute angles that are easier to work with.

A reference angle is always positive and lies between 0° and 90°. It is used to determine the sine, cosine, and tangent of the actual angle by leveraging the corresponding acute angle and its position in the coordinate plane.

Some key points when considering reference angles:

• They help in finding equivalent angle measures when mapped onto a single quadrant.
• They are formed by dropping a perpendicular from the terminal side of the angle to the x-axis.
• They simplify calculations by allowing us to use common trigonometric values.

Understanding these concepts makes solving problems associated with angles and their trigonometric functions much more manageable.