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Chapter 5: Problem 11

Convert each angle to degrees. \(\frac{\pi}{9}\) radians

Short Answer

Expert verified
The angle \(\frac{\pi}{9}\) radians is equal to 20 degrees.

Step by step solution

01

Write down the given angle in radians

The given angle is \(\frac{\pi}{9}\) radians.
02

Apply the conversion formula

To convert the angle from radians to degrees, we will use the formula: Degrees = Radians 脳 \(\frac{180^掳}{\pi}\)
03

Substitute the given angle into the formula

We are given the angle \(\frac{\pi}{9}\) radians, so we will substitute it into the conversion formula: Degrees = \(\frac{\pi}{9}\) 脳 \(\frac{180^掳}{\pi}\)
04

Simplify

When we substitute the angle into the formula, the \(\pi\) in the numerator and denominator cancels out: Degrees = \(\frac{1}{9}\) 脳 180^掳
05

Calculate the result

Now we just need to multiply \(\frac{1}{9}\) by 180: Degrees = 20^掳 The angle \(\frac{\pi}{9}\) radians is equal to 20 degrees.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle Measurement
When we talk about angles, we are referring to the amount of rotation needed to move from one ray or line to another, usually around a common vertex. This is essential in many fields, including mathematics, physics, and engineering.
  • Angles are measured in two primary units: degrees and radians.
  • Choosing the right unit depends on the context and the precision required.
Degrees and radians each provide a different perspective on the concept of rotation. Understanding how to measure and convert angles is crucial for solving problems in trigonometry and calculus.
Radian
A radian is a unit of angular measurement used in mathematics. It is based on the radius of a circle. One radian is the angle created when the radius is laid along the circle's circumference. This measurement is particularly useful in calculus and trigonometry.
  • Radian measure relates directly to the radius of a circle, offering a natural mathematical unit when dealing with circular motion and wave functions.
  • There are exactly \(2\pi\) radians in a full circle. This is because the circumference of a circle is \(2\pi r\), and a radian measures the angle created by the arc equal to the radius.
Radians make certain calculations smoother and more intuitive, thanks to their direct relationship with the properties of the circle. Interestingly, most mathematical functions use radians for angular inputs by default.
Degree
Degrees are another unit used to measure angles. Unlike radians, degrees segment a circle into 360 equal parts, representing full rotation.
  • Each part of a degree is represented as one of these 360 segments.
  • Degrees are often favored in fields like navigation, surveying, and daily applications, because of their historical and intuitive nature.
Converting angles between radians and degrees helps bridge the gap between theoretical mathematics and practical application. The conversion factor between degrees and radians is essential, as 1 degree is approximately 0.01745 radians. This is why multiplying by \(\frac{180^\circ}{\pi}\) converts radians to degrees.

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