Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Find the exact degree measure of the angle. (a) \(\frac{2 \pi}{3}\) (b) \(\frac{11 \pi}{6}\) (c) \(\frac{3 \pi}{4}\)

Step-by-Step Solution

Answer

(a) 120 degrees, (b) 330 degrees, (c) 135 degrees.

Step 1: Understanding Radian to Degree Conversion

To convert an angle from radians to degrees, use the formula: \[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]This formula arises because there are \(180\) degrees in \(\pi\) radians.

Step 2: Convert \\(\frac{2\pi}{3}\\) to Degrees

Substitute \(\frac{2\pi}{3}\) into the conversion formula:\[\text{Degrees} = \frac{2\pi}{3} \times \frac{180}{\pi}.\]Simplifying gives:\[\text{Degrees} = \frac{2 \times 180}{3} = 120. \]Thus, \(\frac{2\pi}{3}\) radians is \(120\) degrees.

Step 3: Convert \\(\frac{11\pi}{6}\\) to Degrees

Substitute \(\frac{11\pi}{6}\) into the conversion formula:\[\text{Degrees} = \frac{11\pi}{6} \times \frac{180}{\pi}.\]Simplifying gives:\[\text{Degrees} = \frac{11 \times 180}{6} = 330.\]Therefore, \(\frac{11\pi}{6}\) radians is \(330\) degrees.

Step 4: Convert \\(\frac{3\pi}{4}\\) to Degrees

Substitute \(\frac{3\pi}{4}\) into the conversion formula:\[\text{Degrees} = \frac{3\pi}{4} \times \frac{180}{\pi}.\]Simplifying gives:\[\text{Degrees} = \frac{3 \times 180}{4} = 135.\]Hence, \(\frac{3\pi}{4}\) radians is \(135\) degrees.

Key Concepts

Angle Conversion

Converting angles between different measurement systems is an essential skill in both mathematics and physics. The two most common units for measuring angles are degrees and radians. It's important to understand how these units relate to each other to effectively make conversions.
The relationship between radians and degrees is direct: \(180\) degrees is equivalent to \(\pi \) radians. Therefore, if you know an angle measure in one unit, you can readily find its equivalent in the other using the conversion formula.

• To convert from radians to degrees, use the formula: \[\text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]
• To convert from degrees to radians, the formula is: \[\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\]

This straightforward relationship simplifies the conversion process, though practicing with different values helps to reinforce the concept. Working through these conversions is crucial in fields like trigonometry, physics, and engineering.

Radian Measure

Radians provide a different yet concise way to measure angles, rooted in the geometry of circles. A radian is defined as the angle created when the radius of a circle is wrapped along its circumference. More simply, an angle of 1 radian covers an arc length equal to the radius of the circle.
This unit emerges naturally from the circle, which is why it’s frequently used in higher mathematics and calculus. One complete revolution around a circle, or \(360\) degrees, is equal to \(2\pi \) radians. Thus, knowing the measure in radians allows for a seamless interaction with formulas involving circles and periodic functions.

• Radians are often more convenient than degrees for many mathematical computations.
• They provide a direct relationship between the arc and radius, helping in solving complex problems.

Understanding radians can initially feel abstract, but visualizing how a radius fits around a circle helps make this unit more intuitive.

Degree Measure

The concept of degrees as a measure for angles is familiar and intuitive, especially since it's rooted in everyday language and usage—like tracking the time on a clock. A circle is divided into 360 equal parts, called degrees, and this division makes managing and calculating angles straightforward when dealing with various practical tasks.
Degrees are the historical choice for geometric figures and angles in everyday applications due to their simplicity and ease of understanding. This measurement system is likely based on ancient Babylonian mathematics, where the number 360 was favorable for calculations given its divisibility by many other numbers.

• Angles in degrees are expressed in values ranging from 0 to 360.
• They make sense in practical dimensions, making them easy to apply in real-world problems.

Although degrees have been around much longer than radians, they continue to serve a wide array of applications—from simple navigation to complex scientific calculations.

Trigonometry Basics

Trigonometry is the study of relationships between the angles and sides of triangles. Using trigonometry, especially in relation to the circle, requires understanding angle measurements in both degrees and radians. These conversions play a pivotal role in a wide variety of applications, from architecture to astronomy.
Trigonometry relies on the concept that functions such as sine, cosine, and tangent depend on angle sizes. Knowing how to switch between degrees and radians lets you solve problems and understand graphs of trigonometric functions better.

• Simplify calculations by knowing how to efficiently convert angles.
• Deepen understanding of wave functions and oscillations with radians.

Getting comfortable with both methods of measuring angles is fundamental to mastering trigonometry and applying it to real-world situations, such as analyzing mechanical vibrations or waves.