Problem 5 - Technical Mathematics with Calculus

Question

Find the reference angle for each given angle. $$249.3^{\circ}$$

Step-by-Step Solution

Answer

The reference angle for 249.3 degrees is 69.3 degrees.

Step 1: Determine the Quadrant

The angle 249.3 degrees lies in the third quadrant since it is greater than 180 degrees but less than 270 degrees.

Step 2: Calculate the Reference Angle

To find the reference angle for an angle in the third quadrant, subtract the angle from 180 degrees. Calculate the reference angle for 249.3 degrees: Reference Angle = 249.3 degrees - 180 degrees.

Step 3: Find the Result

Perform the subtraction to obtain the reference angle: Reference Angle = 249.3 degrees - 180 degrees = 69.3 degrees. The reference angle is 69.3 degrees.

Key Concepts

Trigonometry and Reference Angles

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. A critical concept within trigonometry is the reference angle, which is the acute angle formed by the x-axis and a terminal side of an angle positioned in the standard position (where the vertex is at the origin and the initial side is on the positive x-axis). Reference angles are always between 0° and 90° and are useful for finding the sine, cosine, and tangent of the original angle since they share the same trigonometric values as their corresponding acute angles.

To find the reference angle for any given angle, one must first determine the angle's location within the four quadrants of a circle. For instance, the exercise provided requires us to find the reference angle for 249.3°. As the solution elaborates, this angle falls within the third quadrant, and we subtract it from 180° to find the reference angle, which in this case is 69.3°. Understanding this process is crucial as it simplifies complex angle measurements into their acute counterparts, making it easier to apply trigonometric functions.

Quadrants of a Circle

The quadrants of a circle are a foundational concept for understanding angle measurements and their reference angles. A circle is divided into four parts by the x-axis and y-axis, resulting in four quadrants, each measuring 90°. The quadrants are numbered counterclockwise from the positive x-axis: the first quadrant (0° to 90°), second quadrant (90° to 180°), third quadrant (180° to 270°), and fourth quadrant (270° to 360°).

Determining the Quadrant

For any given angle, such as the one in the exercise, 249.3°, determining the correct quadrant is essential. This knowledge is fundamental in assessing the appropriate action for finding the reference angle. For example, if an angle is in the third quadrant (like 249.3° in our exercise), the reference angle would be calculated differently compared to an angle in the first quadrant.

Angle Measurement

Angle measurement is a critical part of geometry and trigonometry. Angles are usually measured in degrees (°), where a full circle is 360°. An angle's measure determines its position and span within a circle and subsequently its relationship to the x and y axes when in standard position.

Measuring angles is the initial step to a multitude of concepts in mathematics, including determining reference angles. The way an angle is measured affects its classification into one of the four quadrants, which then dictates how to calculate the reference angle. When an angle, like the one given in the exercise, 249.3°, is measured, it implies a rotation from the positive x-axis. The knowledge of such measurement is then utilized in the process outlined in the exercise to find the reference angle, an essential step for applying trigonometric identities and solving problems.