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Chapter 3: Problem 4

Express the following angles in radians: (a) \(20.0^{\circ}\), (b) \(50.0^{\circ}\), (c) \(100^{\circ}\). Convert the following angles to degrees: (d) \(0.330\) rad, (e) \(2.10 \mathrm{rad},(\mathrm{f}) 7.70 \mathrm{rad}\).

Short Answer

Expert verified
(a) \(\frac{\pi}{9}\) rad, (b) \(\frac{5\pi}{18}\) rad, (c) \(\frac{5\pi}{9}\) rad; (d) ~18.9掳, (e) ~120.3掳, (f) ~441.6掳.

Step by step solution

01

Understanding Degree to Radian Conversion

To convert angles from degrees to radians, we use the formula: \[ \text{radian} = \text{degree} \times \frac{\pi}{180} \]. This formula helps convert any angle measure in degrees to its equivalent in radians.
02

Convert 20.0 Degrees to Radians

For part (a), use the formula to convert:\[ 20.0^{\circ} \times \frac{\pi}{180} = \frac{20.0\pi}{180} = \frac{\pi}{9} \text{ radians}. \]
03

Convert 50.0 Degrees to Radians

For part (b), use the same conversion formula:\[ 50.0^{\circ} \times \frac{\pi}{180} = \frac{50.0\pi}{180} = \frac{5\pi}{18} \text{ radians}. \]
04

Convert 100 Degrees to Radians

For part (c), use the conversion formula:\[ 100^{\circ} \times \frac{\pi}{180} = \frac{100\pi}{180} = \frac{5\pi}{9} \text{ radians}. \]
05

Understanding Radian to Degree Conversion

To convert angles from radians to degrees, use the formula: \[ \text{degree} = \text{radian} \times \frac{180}{\pi} \]. This converts any radian measure to degrees.
06

Convert 0.330 Radians to Degrees

For part (d), apply the conversion formula:\[ 0.330 \times \frac{180}{\pi} \approx 18.909 \text{ degrees}. \]
07

Convert 2.10 Radians to Degrees

For part (e), use the formula to convert:\[ 2.10 \times \frac{180}{\pi} \approx 120.321 \text{ degrees}. \]
08

Convert 7.70 Radians to Degrees

For part (f), use the formula:\[ 7.70 \times \frac{180}{\pi} \approx 441.633 \text{ degrees}. \]

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Key Concepts

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

Degrees to Radians Conversion
When we talk about converting angles, degrees and radians are two common units used to measure angles. Converting degrees to radians involves a simple formula: \[ \text{radian} = \text{degree} \times \frac{\pi}{180} \].
This formula helps transition an angle from the degree scale to the radian scale, which is often used in mathematics and sciences like physics and engineering.
  • Example: For an angle of 20.0 degrees, using the formula we get: \[ 20.0^{\circ} \times \frac{\pi}{180} = \frac{\pi}{9} \text{ radians}. \]
  • A quick tip: The number \( \pi \) (approximately 3.14159) plays a crucial role here, acting as the bridge between the two units.
This conversion is essential for integrating angle measurements into trigonometric functions which predominantly use radians.
Radians to Degrees Conversion
Conversely, sometimes we need to convert radians back to degrees, especially in contexts where angle measures are communicated in familiar terms. The conversion formula used is:\[ \text{degree} = \text{radian} \times \frac{180}{\pi} \].
This allows us to express the angle in a context more common in everyday life, such as navigation or carpentry.
  • Example: For 0.330 radians, applying the formula gives: \[ 0.330 \times \frac{180}{\pi} \approx 18.909 \text{ degrees}. \]
  • Remember that converting back to degrees can make it easier to mentally visualize the angle in real-world scenarios.
It's important because degrees are a more intuitive measurement scale for most, with a circle being 360 degrees.
Trigonometry
Trigonometry is the branch of mathematics dealing with the relationships between the sides and angles of triangles. It heavily relies on angle measurement, and often uses the radian measure for its calculations. In trigonometry, angles in radians can simplify formulae and calculations, especially those involving periodic functions like sine and cosine.

Using radians rather than degrees simplifies many mathematical equations, particularly in calculus and advance mathematical modeling.
  • The sine, cosine, and tangent functions are key components that rely on radian measures to maintain consistent behaviors and meaningful results in calculus.
  • Unlike degrees, radians directly relate to the arc length on a circle, making them more fitting for various mathematical disciplines.
Understanding both degree and radian measurement is fundamental for solving trigonometric problems effectively.
Angle Measurement
Angle measurement can be done in degrees or radians. Both are valid ways to quantify angles, whether in a simple geometrical figure or more complex trigonometric problems.
  • Degrees: Breaking down a circle into 360 equal parts, degrees offer a straightforward and familiar means of intangible measurement.
  • Radians: Provide a naturally occurring measurement based on the radius of the circle, where one radian is the angle created by wrapping the radius of a circle along its circumference. Since a full circle is \(2\pi\) radians, radian measures fall elegantly into formulas involving circles and cycles.
When tackling angle measurement, it's crucial to choose the unit that simplifies the problem at hand, ensuring accurate and manageable calculations.

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Most popular questions from this chapter

A particle undergoes three successive displacements in plane, as follows: \(\vec{d}_{1}, 4.00 \mathrm{~m}\) southwest; then \(\vec{d}_{2}, 5.00 \mathrm{~m}\) east; an finally \(\vec{d}_{3}, 6.00 \mathrm{~m}\) in a direction \(60.0^{\circ}\) north of east. Choose a coon dinate system with the \(y\) axis pointing north and the \(x\) axis pointin east. What are (a) the \(x\) component and (b) the \(y\) component of \(\vec{d}_{1}\) What are (c) the \(x\) component and (d) the \(y\) component of \(\vec{d}_{2}\) What are (e) the \(x\) component and (f) the \(y\) component of \(\vec{d}_{3}\) Next, consider the net displacement of the particle for the thre successive displacements. What are (g) the \(x\) component, (h) the component, (i) the magnitude, and (j) the direction of the net dis placement? If the particle is to return directly to the starting poin (k) how far and (1) in what direction should it move?

Consider two displacements, one of magnitude \(3 \mathrm{~m}\) and another of magnitude \(4 \mathrm{~m}\). Show how the displacement vectors may be combined to get a resultant displacement of magnitude (a) \(7 \mathrm{~m}\), (b) \(1 \mathrm{~m}\), and (c) \(5 \mathrm{~m}\).

Three vectors are given by \(\vec{a}=3.0 \hat{\mathrm{i}}+3.0 \hat{\mathrm{j}}-2.0 \hat{\mathrm{k}}\), \(\vec{b}=-1.0 \hat{\mathrm{i}}-4.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}}\), and \(\vec{c}=2.0 \hat{\mathrm{i}}+2.0 \hat{\mathrm{j}}+1.0 \hat{\mathrm{k}}\). Find (a) \(\vec{a} \cdot(\vec{b} \times \vec{c})\), (b) \(\vec{a} \cdot(\vec{b}+\vec{c})\), and \((\mathrm{c}) \vec{a} \times(\vec{b}+\vec{c})\)

Two vectors, \(\vec{r}\) and \(\vec{s}\), lie in the \(x y\) plane. Their magnitudes are \(4.50\) and \(7.30\) units, respectively, and their directions are \(320^{\circ}\) and \(85.0^{\circ}\), respectively, as measured counterclockwise from the positive \(x\) axis. What are the values of (a) \(\vec{r} \cdot \vec{s}\) and (b) \(\vec{r} \times \vec{s}\) ?

You are to make four straight-line moves over a flat desert floor, starting at the origin of an \(x y\) coordinate system and ending at the \(x y\) coordinates \((-140 \mathrm{~m}, 30 \mathrm{~m})\). The \(x\) component and \(y\) component of your moves are the following, respectively, in meters: \((20\) and 60\()\), then \(\left(b_{x}\right.\) and \(\left.-70\right)\), then \(\left(-20\right.\) and \(\left.c_{y}\right)\), then \((-60\) and \(-70\) ). What are (a) component \(b_{x}\) and (b) component \(c_{y}\) ? What are (c) the magnitude and (d) the angle (relative to the positive direction of the \(x\) axis) of the overall displacement?
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