Problem 13 - Precalculus: Functions and Graphs

Question

Exer. \(13-16:\) Find the exact degree measure of the angle. (a) \(\frac{2 \pi}{3}\) (b) \(\frac{11 \pi}{6} \quad\) (c) \(\frac{3 \pi}{4}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to convert radians to degrees. Recall that one full circle (360 degrees) is equivalent to \(2\pi\) radians. Hence, \(1\) radian is \(\frac{180}{\pi}\) degrees.

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

Multiply \(\frac{2\pi}{3}\) by \(\frac{180}{\pi}\) to convert it to degrees: \[ \frac{2\pi}{3} \times \frac{180}{\pi} = \frac{2 \times 180}{3} = 120 \text{ degrees}. \]

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

Multiply \(\frac{11\pi}{6}\) by \(\frac{180}{\pi}\) to convert it to degrees: \[ \frac{11\pi}{6} \times \frac{180}{\pi} = \frac{11 \times 180}{6} = 330 \text{ degrees}. \]

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

Multiply \(\frac{3\pi}{4}\) by \(\frac{180}{\pi}\) to convert it to degrees: \[ \frac{3\pi}{4} \times \frac{180}{\pi} = \frac{3 \times 180}{4} = 135 \text{ degrees}. \]

Step 5: Conclusion

We have completed converting all three angles from radians to degrees. The results are: (a) \(120\) degrees, (b) \(330\) degrees, (c) \(135\) degrees.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that deals with angles, triangles, and the relationships between their sides and the functions that describe these relationships.
Trigonometry is essential to understanding how angles work, especially in the context of the unit circle. Some basic trigonometric functions are sine, cosine, and tangent. They are used to determine the relationships between the angles and sides of a right triangle.
An important concept in trigonometry is the conversion between radians and degrees, which allows mathematicians and engineers to calculate angles in various forms. Understanding trigonometry opens up a whole world of applications in physics, engineering, and even computer graphics. You'll encounter trigonometry in fields like these whenever you need to analyze or simulate real-world phenomena.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It is a helpful tool in trigonometry for exploring angles and their measures.
In the unit circle, angles can be measured in both degrees and radians.
This makes it a crucial concept when dealing with problems that involve angle measurements. Each point on the unit circle corresponds to an angle formed with the positive x-axis. These angles help define the trigonometric functions:

• Sine is the y-coordinate of a point on the unit circle.
• Cosine is the x-coordinate of a point on the unit circle.
• Tangent is the ratio of sine to cosine.

The unit circle allows for a unified way to understand trigonometric functions, making complex calculations more straightforward.

Angle Measurement

Angle measurement is a critical concept in not only trigonometry but also in many other sciences and engineering fields. Angles can be measured in degrees or radians.
Most people are familiar with degrees, where a full circle is split into 360 parts. However, radians offer a more mathematical way to express angles.Radians are based on the radius of the circle:

• One full circle, or 360 degrees, equals \( 2\pi \) radians.
• Therefore, 1 radian is equivalent to \(\frac{180}{\pi}\) degrees.

By understanding how to convert between these two units—like we did in the original exercise—you'll be better equipped to tackle problems in physics and engineering where radians give a more natural frame of reference.
This makes computations, such as those involving periodic functions, more intuitive and applicable to real-world scenarios.