Compare An Angle Having A Measure Of $120^{\circ}$ With That Of An Angle Whose Measure Is $\frac{5 \pi}{6}$ Radians. Explain Your Reasoning.

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Introduction

Angles are a fundamental concept in mathematics, and they can be measured in different units, such as degrees and radians. In this article, we will compare an angle with a measure of $120^{\circ}$ with that of an angle whose measure is $\frac{5 \pi}{6}$ radians. We will explain our reasoning and provide a step-by-step guide to help you understand the relationship between these two angles.

Understanding Degrees and Radians

Before we dive into the comparison, let's briefly review the concepts of degrees and radians.

  • Degrees: A degree is a unit of measurement for angles, where a full circle is equal to 360 degrees. Angles can be measured in degrees, and they are commonly used in everyday applications, such as navigation and architecture.
  • Radians: A radian is a unit of measurement for angles, where a full circle is equal to $2 \pi$ radians. Radians are commonly used in mathematics and physics, particularly in trigonometry and calculus.

Converting Degrees to Radians

To compare the two angles, we need to convert the angle with a measure of $120^{\circ}$ to radians. We can use the following formula to convert degrees to radians:

radians=degrees×π180\text{radians} = \text{degrees} \times \frac{\pi}{180}

Plugging in the value of $120^{\circ}$, we get:

radians=120×π180=2π3\text{radians} = 120 \times \frac{\pi}{180} = \frac{2 \pi}{3}

Comparing the Angles

Now that we have converted the angle with a measure of $120^{\circ}$ to radians, we can compare it with the angle whose measure is $\frac{5 \pi}{6}$ radians.

  • Angle 1: $\frac{2 \pi}{3}$ radians (converted from $120^{\circ}$)
  • Angle 2: $\frac{5 \pi}{6}$ radians

To compare these two angles, we can use the following approach:

  1. Find the common denominator: We need to find a common denominator for the two fractions. In this case, the common denominator is 6.
  2. Convert both fractions to have the common denominator: We can convert both fractions to have a denominator of 6 by multiplying the numerator and denominator of each fraction by the necessary factor.
  3. Compare the numerators: Once we have both fractions with the same denominator, we can compare the numerators to determine which angle is larger.

Step-by-Step Comparison

Let's follow the steps outlined above to compare the two angles:

  1. Find the common denominator: The common denominator for the two fractions is 6.
  2. Convert both fractions to have the common denominator: We can convert both fractions to have a denominator of 6 by multiplying the numerator and denominator of each fraction by the necessary factor.

For Angle 1:

2π3=2π×23×2=4π6\frac{2 \pi}{3} = \frac{2 \pi \times 2}{3 \times 2} = \frac{4 \pi}{6}

For Angle 2:

\frac{5 \pi}{6}$ (no conversion needed) 3. **Compare the numerators**: Now that we have both fractions with the same denominator, we can compare the numerators to determine which angle is larger. $\frac{4 \pi}{6} < \frac{5 \pi}{6}

Therefore, the angle with a measure of $\frac{5 \pi}{6}$ radians is larger than the angle with a measure of $120^{\circ}$.

Conclusion

In this article, we compared an angle with a measure of $120^{\circ}$ with that of an angle whose measure is $\frac{5 \pi}{6}$ radians. We explained our reasoning and provided a step-by-step guide to help you understand the relationship between these two angles. By converting the angle with a measure of $120^{\circ}$ to radians and comparing the two angles, we determined that the angle with a measure of $\frac{5 \pi}{6}$ radians is larger.

Key Takeaways

  • Angles can be measured in different units, such as degrees and radians.
  • To compare angles in different units, we need to convert them to a common unit.
  • The formula to convert degrees to radians is: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$
  • To compare two angles, we can find the common denominator, convert both fractions to have the common denominator, and compare the numerators.

Frequently Asked Questions

  • Q: What is the difference between degrees and radians? A: Degrees and radians are two different units of measurement for angles. A full circle is equal to 360 degrees, while a full circle is equal to $2 \pi$ radians.
  • Q: How do I convert degrees to radians? A: You can use the formula: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$
  • Q: How do I compare two angles in different units? A: You can find the common denominator, convert both fractions to have the common denominator, and compare the numerators.
    Frequently Asked Questions: Angles in Degrees and Radians =============================================================

Introduction

In our previous article, we compared an angle with a measure of $120^{\circ}$ with that of an angle whose measure is $\frac{5 \pi}{6}$ radians. We explained our reasoning and provided a step-by-step guide to help you understand the relationship between these two angles. In this article, we will answer some frequently asked questions about angles in degrees and radians.

Q&A

Q: What is the difference between degrees and radians?

A: Degrees and radians are two different units of measurement for angles. A full circle is equal to 360 degrees, while a full circle is equal to $2 \pi$ radians.

Q: How do I convert degrees to radians?

A: You can use the formula: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

Q: How do I convert radians to degrees?

A: You can use the formula: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Q: What is the relationship between degrees and radians?

A: The relationship between degrees and radians is that they are two different units of measurement for angles. A full circle is equal to 360 degrees, while a full circle is equal to $2 \pi$ radians. This means that there are $\frac{2 \pi}{360} = \frac{\pi}{180}$ radians in one degree.

Q: How do I compare two angles in different units?

A: You can find the common denominator, convert both fractions to have the common denominator, and compare the numerators.

Q: What is the common denominator for degrees and radians?

A: The common denominator for degrees and radians is 1.

Q: How do I find the common denominator for two fractions?

A: You can find the common denominator by finding the least common multiple (LCM) of the two denominators.

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

Q: How do I find the LCM of two numbers?

A: You can find the LCM of two numbers by listing the multiples of each number and finding the smallest number that appears in both lists.

Q: What is the relationship between the LCM and the GCD?

A: The relationship between the LCM and the GCD (greatest common divisor) is that the product of the LCM and the GCD is equal to the product of the two numbers.

Q: How do I find the GCD of two numbers?

A: You can find the GCD of two numbers by listing the factors of each number and finding the largest factor that appears in both lists.

Q: What is the relationship between the GCD and the LCM?

A: The relationship between the GCD and the LCM is that the product of the GCD and the LCM is equal to the product of the two numbers.

Conclusion

In this article, we answered some frequently asked questions about angles in degrees and radians. We explained the difference between degrees and radians, how to convert between the two units, and how to compare two angles in different units. We also discussed the relationship between the least common multiple (LCM) and the greatest common divisor (GCD). We hope that this article has been helpful in answering your questions about angles in degrees and radians.

Key Takeaways

  • Degrees and radians are two different units of measurement for angles.
  • To convert degrees to radians, use the formula: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$
  • To convert radians to degrees, use the formula: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$
  • To compare two angles in different units, find the common denominator, convert both fractions to have the common denominator, and compare the numerators.
  • The relationship between the LCM and the GCD is that the product of the LCM and the GCD is equal to the product of the two numbers.
  • The relationship between the GCD and the LCM is that the product of the GCD and the LCM is equal to the product of the two numbers.