Problem 37 - Algebra and Trigonometry

Question

find the reference angle for each angle. $$ 205^{\circ} $$

Step-by-Step Solution

Answer

The reference angle for \(205^{\circ}\) is \(25^{\circ}\)

Step 1: Identify the Quadrant of the Angle

First, we need to establish in which quadrant the angle falls. Since 205 degrees is greater than 180 degrees but less than 270 degrees, it falls in the third quadrant.

Step 2: Subtract from 180 degrees

Angles in the third quadrant have reference angles equal to the absolute difference from 180 degrees. Therefore, the reference angle for 205 degrees in the third quadrant is given by: \[ \textit{reference angle} = |\textit{given angle} - 180^{\circ}|\]

Step 3: Calculation

Using the above formula, calculate: \[ |\textit{reference angle}| = |205^{\circ} - 180^{\circ}| = |25^{\circ}| = 25^{\circ}\]

Key Concepts

Trigonometry

Trigonometry is a fascinating branch of mathematics that deals with the relationships between the sides and angles of triangles. The word 'trigonometry' itself comes from two Greek terms, 'trigonon' meaning triangle, and 'metron' meaning measure. This subject is essential not only in theoretical mathematical problems but also in applied sciences and various real-world applications like architecture, engineering, and even astronomy.

One key aspect of trigonometry is the study of reference angles, which are used to simplify the process of finding the sine, cosine, and tangent values of larger angles. A reference angle is always the acute angle (less than 90 degrees) that a given angle makes with the x-axis, and it can be found for any angle in all four quadrants of the Cartesian plane. The concept of the reference angle is particularly useful because it allows us to use the properties of right triangles to solve problems involving any angle, no matter how large.

Quadrants of an Angle

The quadrants of an angle refer to the division of the coordinate plane into four regions by the x and y axes, with each quadrant containing a range of angles with specific characteristics. These quadrants are labeled as follows:

• Quadrant I: Angles from 0 to 90 degrees (0 to \(\frac{\pi}{2}\) radians)
• Quadrant II: Angles from 90 to 180 degrees (\(\frac{\pi}{2}\) to \(\pi\) radians)
• Quadrant III: Angles from 180 to 270 degrees (\(\pi\) to \(\frac{3\pi}{2}\) radians)
• Quadrant IV: Angles from 270 to 360 degrees (\(\frac{3\pi}{2}\) to \(2\pi\) radians)

Identifying the quadrant in which an angle resides is crucial because it determines the sign (positive or negative) of the trigonometric functions sine, cosine, and tangent for that angle. Moreover, it aids in finding the reference angle, which is always positive and measured relative to the x-axis.

Angle Measurement

Angle measurement is a fundamental concept in geometry, where an angle is defined as the figure formed by two rays, called the sides of the angle, sharing a common endpoint, known as the vertex. Angles can be measured in degrees or radians. A complete revolution is equal to 360 degrees or \(2\pi\) radians, and the degree system historically comes from the Babylonians, who used a base-60 numerical system.

When measuring angles, it's important to start from the positive x-axis and measure counterclockwise. This is considered a positive angle measurement. If we measure clockwise, the angle is said to be negative. Understanding how to measure angles is vital when applying the knowledge to find reference angles, which in turn simplifies the trigonometric calculations for any angle, regardless of the quadrant it may fall into.