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Chapter 4: Problem 4

Convert the given angle to radians. \(275^{\circ}\)

Short Answer

Expert verified
\( \frac{55 \text{蟺}}{36} \text{ radians} \)

Step by step solution

01

Understand the Conversion Formula

The formula to convert degrees to radians is: \[ \text{radians} = \text{degrees} \times \frac{\text{蟺}}{180} \]
02

Identify the given angle in degrees

The given angle is \(275^{\text{掳}}\).
03

Set up the conversion

Substitute \(275^{\text{掳}}\) into the formula: \[ \text{radians} = 275 \times \frac{\text{蟺}}{180} \]
04

Simplify the expression

Simplify the fraction: \[ 275 \times \frac{\text{蟺}}{180} = \frac{275\text{蟺}}{180} \]
05

Reduce the fraction to simplest form

Divide both the numerator and the denominator by their greatest common divisor (5): \[ \frac{275 \text{蟺}}{180} = \frac{55 \text{蟺}}{36} \]

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Key Concepts

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

Understanding Angle Conversion
Angles can be measured in degrees or radians. It's important to know both units because they are used in different contexts in mathematics and physics. Degrees are more common in everyday use, while radians are often used in higher-level mathematics, like calculus.

Converting from degrees to radians involves a simple formula. The key to understanding this conversion is knowing that 蟺 (pi) radians is equal to 180 degrees. This relationship forms the basis for the conversion formula.
Using the Radians Formula
To convert degrees to radians, you use the formula: \[ \text{radians} = \text{degrees} \times \frac{\text{蟺}}{180} \]

This formula arises from the proportion of the circle's angle in degrees to the same angle in radians. For instance, if you need to convert 275 degrees to radians, you plug 275 into the formula:

\[ \text{radians} = 275 \times \frac{\text{蟺}}{180} \]

By multiplying, you get the angle in radians. This is an essential skill for solving many problems in trigonometry and calculus.
Simplifying Fractions in Angle Conversion
Once you plug the degrees into the radians formula, you often get a fraction that can be simplified. Simplifying fractions makes your answer cleaner and easier to understand.

For example, take the previous conversion: \[ 275 \times \frac{\text{蟺}}{180} = \frac{275\text{蟺}}{180} \]

Here, simplifying is important for clarity. Both 275 and 180 can be divided by 5 (their greatest common divisor), simplifying the fraction:

\[ \frac{275\text{蟺}}{180} = \frac{55\text{蟺}}{36} \]

Always look for the greatest common divisor when reducing fractions. This final step ensures your answer is in its simplest form, making it more elegant and easier to use in further calculations.

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Most popular questions from this chapter

Assume that a particle moves along a circle of radius \(r\) for a period of time \(t\). Given either the arc length \(s\) or the central angle \(\theta\) swept out by the particle, find the linear and angular speed of the particle. \(r=7 \mathrm{~m}, t=3.2\) sec, \(\theta=172^{\circ}\)

Convert the given angle to radians. \(4^{\circ}\)

Assume that a particle moves along a circle of radius \(r\) for a period of time \(t\). Given either the arc length \(s\) or the central angle \(\theta\) swept out by the particle, find the linear and angular speed of the particle. \(r=2 \mathrm{~m}, t=1.6\) sec, \(s=6 \mathrm{~m}\)

Assume that a particle moves along a circle of radius \(r\) for a period of time \(t\). Given either the arc length \(s\) or the central angle \(\theta\) swept out by the particle, find the linear and angular speed of the particle. \(r=1.5 \mathrm{ft}, t=0.3\) sec, \(s=4\) in

Convert the given angle to radians. \(130^{\circ}\)
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