Which Of The Following Are In Correct Order From Greatest To Least?A. Π 2 , 330 ∘ , 5 Π 3 , 7 Π 6 , 2 Π 3 \frac{\pi}{2}, 330^{\circ}, \frac{5 \pi}{3}, \frac{7 \pi}{6}, \frac{2 \pi}{3} 2 Π ​ , 33 0 ∘ , 3 5 Π ​ , 6 7 Π ​ , 3 2 Π ​ B. 5 Π 3 , 7 Π 6 , 2 Π 3 , Π 2 , 330 ∘ \frac{5 \pi}{3}, \frac{7 \pi}{6}, \frac{2 \pi}{3}, \frac{\pi}{2}, 330^{\circ} 3 5 Π ​ , 6 7 Π ​ , 3 2 Π ​ , 2 Π ​ , 33 0 ∘ C.

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Introduction

In trigonometry, angles are a fundamental concept that plays a crucial role in solving various mathematical problems. Angles can be measured in degrees or radians, and it's essential to understand the relationship between these two units of measurement. In this article, we will explore the given options and determine which one is in the correct order from greatest to least.

Understanding the Options

Before we dive into the analysis, let's understand the given options:

  • Option A: π2,330∘,5π3,7π6,2π3\frac{\pi}{2}, 330^{\circ}, \frac{5 \pi}{3}, \frac{7 \pi}{6}, \frac{2 \pi}{3}
  • Option B: 5π3,7π6,2π3,π2,330∘\frac{5 \pi}{3}, \frac{7 \pi}{6}, \frac{2 \pi}{3}, \frac{\pi}{2}, 330^{\circ}
  • Option C: (Not provided)

Converting Angles to a Common Unit

To compare the angles, we need to convert them to a common unit. Let's convert all the angles to radians:

  • π2\frac{\pi}{2} is already in radians.
  • 330∘=330π180=11π6330^{\circ} = \frac{330 \pi}{180} = \frac{11 \pi}{6} radians.
  • 5π3\frac{5 \pi}{3} is already in radians.
  • 7π6\frac{7 \pi}{6} is already in radians.
  • 2π3\frac{2 \pi}{3} is already in radians.

Comparing the Angles

Now that we have all the angles in radians, let's compare them:

  • π2=1.57081\frac{\pi}{2} = \frac{1.5708}{1}
  • 5π3=5.23501\frac{5 \pi}{3} = \frac{5.2350}{1}
  • 7π6=3.66501\frac{7 \pi}{6} = \frac{3.6650}{1}
  • 2π3=2.09441\frac{2 \pi}{3} = \frac{2.0944}{1}
  • 11π6=5.78501\frac{11 \pi}{6} = \frac{5.7850}{1}

Determining the Correct Order

Based on the comparison, we can see that the correct order from greatest to least is:

  • 11π6\frac{11 \pi}{6}
  • 5π3\frac{5 \pi}{3}
  • 7π6\frac{7 \pi}{6}
  • 2π3\frac{2 \pi}{3}
  • π2\frac{\pi}{2}

Conclusion

In conclusion, the correct order from greatest to least is Option B: 5π3,7π6,2π3,π2,330∘\frac{5 \pi}{3}, \frac{7 \pi}{6}, \frac{2 \pi}{3}, \frac{\pi}{2}, 330^{\circ}. This order makes sense because 11π6\frac{11 \pi}{6} is the largest angle, followed by 5π3\frac{5 \pi}{3}, then 7π6\frac{7 \pi}{6}, and so on.

Final Thoughts

Introduction

In our previous article, we explored the concept of angles in trigonometry and determined the correct order from greatest to least. In this article, we will address some of the most frequently asked questions related to angles in trigonometry.

Q: What is the difference between degrees and radians?

A: Degrees and radians are two units of measurement for angles. Degrees are commonly used in everyday life, while radians are used in mathematical calculations. To convert degrees to radians, we multiply the angle in degrees by π180\frac{\pi}{180}.

Q: How do I convert an angle from degrees to radians?

A: To convert an angle from degrees to radians, we use the formula:

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

For example, to convert 30∘30^{\circ} to radians, we multiply 3030 by π180\frac{\pi}{180}:

30∘×π180=π630^{\circ} \times \frac{\pi}{180} = \frac{\pi}{6}

Q: How do I convert an angle from radians to degrees?

A: To convert an angle from radians to degrees, we use the formula:

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

For example, to convert π6\frac{\pi}{6} to degrees, we multiply π6\frac{\pi}{6} by 180π\frac{180}{\pi}:

π6×180π=30∘\frac{\pi}{6} \times \frac{180}{\pi} = 30^{\circ}

Q: What is the relationship between sine, cosine, and tangent?

A: Sine, cosine, and tangent are three fundamental trigonometric functions that are related to each other. The sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. The cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Q: How do I use the unit circle to find trigonometric values?

A: The unit circle is a powerful tool for finding trigonometric values. The unit circle is a circle with a radius of 1 that is centered at the origin of the coordinate plane. By using the unit circle, we can find the values of sine, cosine, and tangent for any angle.

Q: What is the difference between a radian and a degree?

A: A radian is a unit of measurement for angles that is based on the ratio of the arc length to the radius of a circle. A degree is a unit of measurement for angles that is based on a 360-degree circle. While both units are used to measure angles, they are not equivalent.

Conclusion

In conclusion, understanding angles in trigonometry is crucial for solving various mathematical problems. By addressing some of the most frequently asked questions related to angles in trigonometry, we can gain a deeper understanding of this fundamental concept. Whether you are a student or a professional, having a solid grasp of angles in trigonometry can help you tackle complex problems with confidence.

Additional Resources

For further learning, we recommend the following resources:

  • Khan Academy: Trigonometry
  • MIT OpenCourseWare: Trigonometry
  • Wolfram MathWorld: Trigonometry

By exploring these resources, you can gain a deeper understanding of angles in trigonometry and improve your problem-solving skills.