Problem 15 - Algebra and Trigonometry

Question

Find the degree measure of the angle with the given radian measure. $$ \frac{7 \pi}{6} $$

Step-by-Step Solution

Answer

\( \frac{7\pi}{6} \) radians is 210 degrees.

Step 1: Understand the Relationship between Radians and Degrees

Angles can be measured in two units: radians and degrees. To convert an angle from radians to degrees, we use the conversion factor \( 180^\circ / \pi \text{ radians} \). This means that 1 radian is equivalent to approximately 57.2958 degrees.

Step 2: Setup the Conversion Formula

The formula to convert radians to degrees is: \[\text{Degrees} = \text{Radians} \times \left( \frac{180^\circ}{\pi} \right)\]We'll use this formula to convert \( \frac{7\pi}{6} \) radians to degrees.

Step 3: Substitute and Simplify

Substitute \( \frac{7\pi}{6} \) into the conversion formula: \[\text{Degrees} = \frac{7\pi}{6} \times \left( \frac{180^\circ}{\pi} \right)\]Cancel out \( \pi \) in the numerator and the denominator, and then simplify: \[\text{Degrees} = \frac{7 \times 180^\circ}{6}\]

Step 4: Perform the Multiplication and Division

Calculate the multiplication and division: \[\text{Degrees} = \frac{1260}{6} = 210^\circ\]Thus, \( \frac{7\pi}{6} \) radians is equivalent to 210 degrees.

Key Concepts

Understanding Degree Measure

Degree measure is a way of expressing angles in a system based on dividing a circle into 360 equal parts. Each part is one degree. This system is commonly used in geometry, navigation, and various applications requiring precise angle calculations.

Here's why degree measure is significant:

• Intuitive Understanding: Degrees provide a more intuitive grasp of angles for most people compared to radians.
• Division of Circle: A full circle is 360 degrees, making it convenient for addressing angles in quadrants and various geometrical shapes.
• Common Usage: Most trigonometric tables and tools are designed with degrees in mind.

Whether you're evaluating the height of a tree using its shadow or plotting a navigation course, understanding degree measure is crucial for accurate calculations.

Mastering Angle Conversion

Converting angles from radians to degrees is a critical skill, especially in math and engineering fields where both systems are used.

Angle conversion involves a straightforward process where you:

• Utilize the formula: Convert using the formula: \[ \text{Degrees} = \text{Radians} \times \left( \frac{180^\circ}{\pi} \right) \]
• Apply Conversion Factor: A conversion factor of \(180^\circ/\pi\) is used because it clarifies the equivalence relationship between these two units.
• Perform Simple Arithmetic: Multiply the radian measure by the conversion factor for the angle in degrees.

When you have an angle given in radians like \(\frac{7\pi}{6}\), you simply multiply by the conversion factor to find its degree measure. This process helps in seamless transitions between these units during problem-solving.

Exploring Trigonometric Concepts

Trigonometric concepts are the foundation of many areas in mathematics and science, revolving around the study of angles and their relationships.

When working with radians and degrees, understanding trigonometrics allows for more versatile applications:

• Sine, Cosine, Tangent: These fundamental trigonometric functions apply to both radians and degrees, enabling calculations of side lengths and angles in triangles.
• Periodic Functions: Trigonometric functions are periodic, with cycles that can be described in both radian and degree measures. Understanding how a cycle relates to 360 degrees or \(2\pi\) radians is essential.
• Real-world Applications: From music theory to architecture, trigonometry finds its place in many practical and theoretical applications.

By deepening your understanding of trigonometric concepts, you can bridge the gap between theoretical angles and their real-world significance, particularly when converting between radians and degrees.