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Chapter 1: Problem 9

You know that a \(360^{\circ}\) rotation is one complete rotation around a circle. Find the degree measures for each of these rotations. a. half a rotation b. two complete rotations c. \(1 \frac{1}{2}\) rotations

Short Answer

Expert verified
a. 180掳 b. 720掳 c. 540掳

Step by step solution

01

Understand the problem

Identify the type of rotation and the degrees associated. Recall that one complete rotation is equivalent to 360 degrees.
02

Calculate half a rotation

For half a rotation, divide 360 degrees by 2: \( 360^\text{掳} \times \frac{1}{2} = 180^\text{掳} \).
03

Calculate two complete rotations

Multiply the degrees of one rotation by 2: \( 360^\text{掳} \times 2 = 720^\text{掳} \).
04

Calculate \(1 \frac{1}{2}\) rotations

First, convert \(1 \frac{1}{2}\) to an improper fraction, which is \( \frac{3}{2} \). Then multiply by 360 degrees: \( 360^\text{掳} \times \frac{3}{2} = 540^\text{掳} \).

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Key Concepts

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

Degree Measure
Degrees are a way to measure angles and rotations. Imagine a full circle: that circle represents 360 degrees.
If you split the circle in half, each half would be 180 degrees.
To measure angles less than a full circle, we use smaller fractions of these 360 degrees.
This is important for understanding how rotations work.
Always remember these key points:
  • One full circle = 360 degrees
  • Half a circle = 180 degrees
  • Quarter of a circle = 90 degrees
Knowing how to divide and multiply these degrees lets us calculate rotations easily.
Complete Rotation
A complete rotation means moving around a full circle, starting and ending at the same position.
In degrees, this is always 360 degrees.
This concept is essential in everyday life and various applications:
  • Clocks: The hour hand makes a full rotation every 12 hours.
  • Wheels: When a car wheel turns once, it makes a complete rotation.
  • Geometry: Understanding full rotations helps in solving various geometric problems.
It's also critical in our exercise: calculating multiple and partial rotations starts with knowing a full rotation.
Fractional Rotation
A fractional rotation is any rotation that is less than a full 360 degrees or more than a single full rotation if it is repetitive.
For example, if you only turn halfway around, that's half a rotation or 180 degrees.
Here's how fractional rotations work in our examples:
  • Half a rotation: This is \(\frac{1}{2}\) of 360 degrees. \(360^{\text{掳}} \times \frac{1}{2} = 180^{\text{掳}}\).
  • Two complete rotations: This is \(\frac{2}{1}\) times a full 360 degrees. \(360^{\text{掳}} \times 2 = 720^{\text{掳}}\).
  • One and a half rotations: This is \(\frac{3}{2}\) times a full 360 degrees. \(360^{\text{掳}} \times \frac{3}{2} = 540^{\text{掳}}\).
Understanding fractional rotations lets you solve more complex problems by breaking them down into smaller parts.
It becomes very straightforward once you think of it as multiplying a fraction by 360 degrees.

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Most popular questions from this chapter

Find three angles in your home or school with measures equal to \(90^{\circ}\) three with measures less than \(90^{\circ},\) and three with measures greater than \(90^{\circ} .\) Describe where you found each angle.

Tell whether each expression was evaluated correctly using the order of operations. If not, give the correct result. \(54-27 \div 3=45\)

\(A\) \(180^{\circ}\) angle is sometimes called a straight angle. Explain why that name makes sense.

Write thirty-two thousand, five hundred sixty-three in standard form.

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