Problem 4 - Technical Mathematics with Calculus

Question

Find the reference angle for each given angle. $$138.6^{\circ}$$

Step-by-Step Solution

Answer

The reference angle for 138.6 degrees is 41.4 degrees.

Step 1: Determine the quadrant of the given angle

First, determine in which quadrant the given angle lies. The given angle is 138.6 degrees which lies in the second quadrant (between 90 and 180 degrees).

Step 2: Find the reference angle for the second quadrant

To find the reference angle for an angle in the second quadrant, subtract the angle from 180 degrees: \[ 180^\circ - 138.6^\circ = 41.4^\circ. \]

Step 3: State the reference angle

The reference angle for an angle in the second quadrant is the absolute difference between the angle and 180 degrees. Therefore, the reference angle for 138.6 degrees is 41.4 degrees.

Key Concepts

Quadrants of Angles

Understanding the quadrants of angles is crucial in trigonometry. The coordinates plane is divided into four sections called quadrants. These are ordered counterclockwise and numbered from I to IV.

Each quadrant represents a range of angles:

• Quadrant I: 0 to 90 degrees
• Quadrant II: 90 to 180 degrees
• Quadrant III: 180 to 270 degrees
• Quadrant IV: 270 to 360 degrees

Knowing which quadrant an angle falls in determines what the signs of the trigonometric functions will be and helps guide us in finding the reference angle, a concept that simplifies working with trigonometric ratios across all quadrants.

Measuring Angles

Measuring angles is a foundational skill in mathematics, especially in the study of geometry and trigonometry. The unit typically used is degrees or radians.

Here’s how angles are measured:

• A full circle is 360 degrees or \( 2\pi \) radians.
• A right angle is 90 degrees or \( \frac{\pi}{2} \) radians.
• Acute angles are less than 90 degrees.
• Obtuse angles are more than 90 degrees but less than 180 degrees.

Angles guide us in determining shapes, making geometric calculations, and understanding the position of different lines and planes relative to one another. Calculating a reference angle always involves measuring the acute angle between the terminal side of the given angle and the horizontal axis.

Trigonometric Concepts

Trigonometric concepts involve the study and application of trigonometric functions—sine, cosine, and tangent— which are fundamental in relating the angles of a triangle to the lengths of its sides. They extended to non-right angled triangles through laws of sines and cosines, and even to periodic phenomena, waves, and oscillations.

Understanding trigonometric concepts is not only essential for solving geometry problems but also for various real-world applications such as engineering, physics, and even in certain fields of economics and medical imaging. The concept of a reference angle allows us to use trigonometric functions for any angle by determining its corresponding acute angle within the first quadrant, where these functions are most easily defined and understood.