Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Convert each radian measure to degrees. $$\frac{11 \pi}{6}$$

Step-by-Step Solution

Answer

\(\frac{11\pi}{6}\) radians is equivalent to 330 degrees.

Step 1: Understand the Radian to Degree Conversion Formula

To convert radians to degrees, we use the formula: \[\text{degrees} = \text{radians} \times \frac{180}{\pi}\] This formula works because \( \pi \) radians is equivalent to 180 degrees.

Step 2: Apply the Formula to Given Radians

Using the formula from Step 1, substitute \( \frac{11\pi}{6} \) for radians:\[\text{degrees} = \frac{11\pi}{6} \times \frac{180}{\pi}\]

Step 3: Simplify the Expression

Cancel out \( \pi \) in the numerator and denominator:\[\text{degrees} = \frac{11}{6} \times 180\]

Step 4: Complete the Multiplication

Now, multiply \( \frac{11}{6} \) by 180:\[\frac{11 \times 180}{6} \] Divide 180 by 6 to simplify the calculation:\[\frac{11 \times 30}{1} = 330\]

Step 5: State the Final Answer

The conversion of \( \frac{11\pi}{6} \) radians to degrees gives us 330 degrees.

Key Concepts

Trigonometry

Trigonometry is a fascinating branch of mathematics focused on the relationships between angles and the sides of triangles. Its roots can be traced back to the study of astronomy and the need to calculate distances and angles in the sky. It is essential in understanding how to navigate and calculate angles across different disciplines. One important aspect of trigonometry is learning how to convert between different units of angle measurement, such as radians and degrees.

Trigonometry functions like sine, cosine, and tangent play a crucial role in linking angles and side lengths in triangles. These functions depend on angle measures, which can be expressed in both radians and degrees.

• Sine (\(\sin\)
• Cosine (\(\cos\)
• Tangent (\(\tan\)
• Each applies to right-angled triangles and extends to more complex shapes.

Being equipped with these concepts helps solve real-world problems and lays the foundation for advanced fields such as physics and engineering.

Angle Measurement

Angle measurement provides the foundation for discussing the size of an angle in mathematics and other applied sciences. Two primary units are commonly used to measure angles: radians and degrees. While degrees are familiar in everyday life, used in geometry class and navigation, radians offer a more natural measure in advanced mathematics and physics.

The fundamental conversion factor between these two units is \(\pi\) radians equaling 180 degrees. This foundation allows for seamless conversion, vital as different problems require different units for convenience or accuracy.

• Degrees are based on dividing one full circle into 360 units.
• Radians are based on the radius of the circle, where one full circle equals \(2\pi\) radians.

Understanding these concepts is essential when exploring advanced geometry or calculus. The ability to convert between radians and degrees means one can smoothly transition between theory and practical applications.

Mathematics

Mathematics is not just numbers, and calculations but is also about understanding the patterns and relationships that represent fundamental truths. When converting angles from radians to degrees, this illustrates a deeper capability within mathematics to bridge different realms of understanding.

The formula \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\) is a perfect example of mathematical efficiency and elegance. It showcases how the concepts blend seamlessly:

• The value 180 stems from the degrees in a semicircle.
• \(\pi\) is central to describing circular motion and geometry.

Learning mathematical conversion enhances many problem-solving skills, whether it’s handling trigonometric problems or complex integrals. Students find that as they master these concepts, they not only solve problems but also gain insights into how different mathematical theories interconnect.