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Chapter 10: Problem 43

Convert the angle from radian measure into degree measure. $$ -\frac{\pi}{6} $$

Short Answer

Expert verified
The angle \\( -\frac{\pi}{6} \\) in degrees is \\( -30^\circ \\).

Step by step solution

01

Understand the Relationship

To convert an angle from radians to degrees, you need to know that \( \pi \ \text{radians} = 180^\circ \). This relationship will help us convert the given angle.
02

Set Up the Conversion

To convert \( -\frac{\pi}{6} \) to degrees, multiply by the conversion factor, which is \( \frac{180^\circ}{\pi} \). This factor allows us to cancel out \( \pi \) and convert into degrees.
03

Perform the Multiplication

Compute the product: \[-\frac{\pi}{6} \times \frac{180}{\pi} = -\frac{1}{6} \times 180 = -30\]The \( \pi \) in the numerator and denominator cancel out, leaving you with \( -\frac{180}{6} \).
04

Simplify the Expression

Simplify \( -\frac{180}{6} \), which results in \( -30 \). This is the angle in degrees.

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Key Concepts

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

Radian to Degree Conversion
Converting angles from radians to degrees is a vital skill in mathematics, particularly in trigonometry. You'll often need to switch between these two units of measure. The key is using the relationship between radians and degrees:
  • One full circle in radians is expressed as \( 2\pi \; ext{radians}\), and in degrees as \(360^\circ\).
  • Therefore, \(\pi\; ext{radians} = 180^\circ\).
This allows us to derive a conversion factor. For any radian measure \(x\), to convert it to degrees, multiply it by \(\frac{180^\circ}{\pi}\). This factor essentially converts the radian to its degree equivalent by making use of the \(\pi\) and \(180^\circ\) relation.
Angle Measure
Angles are measured in different units, with radians and degrees being the most common. Understanding angle measurement is fundamental in geometry and trigonometry.
  • Degrees are based on dividing a circle into 360 equal parts. This system is ancient and widely used in various practical applications.
  • Radians are a bit more abstract and relate the angle's measure to the radius of a circle. It's defined such that the angle formed by an arc length equal to the radius is 1 radian.
When working with angles, it's important to understand which unit to use, depending on the calculations or applications you are working with.
Trigonometry
Trigonometry is a branch of mathematics that examines the relationships between the sides and angles of triangles. In trigonometry, angle conversion between radians and degrees is key.
  • The sine, cosine, and tangent functions are dependent on the angle measurement unit.
  • In calculus, trigonometric functions are often expressed in radians due to their natural properties and easier integration and differentiation.
Familiarity with both units ensures you can effectively solve trigonometry problems and comprehend theoretical concepts.
Mathematics
Mathematics is a broad field that spans numerous sub-disciplines, each with its own set of tools and concepts. Angle conversion plays a significant role in many mathematical contexts:
  • In geometry, calculating angles accurately requires understanding the conversion between different units.
  • In physics, angles in radians help express rotational dynamics.
Mastery of this concept not only aids in understanding but also enhances your ability to apply mathematical models correctly across various fields. Mathematics uses these principles to build on simple numbers and geometry to explore complex theories and solve practical problems.

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Most popular questions from this chapter

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(-\frac{\pi}{2}\right)\right)\)

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \pi\right]\) and that the range of arccosecant is \(\left[-\frac{\pi}{2}, 0\right) \cup\left(0, \frac{\pi}{2}\right]\) when finding the exact value. $$ \operatorname{arcsec}\left(\sec \left(\frac{5 \pi}{6}\right)\right) $$

Assume that the range of arcsecant is \(\left[0, \frac{\pi}{2}\right) \cup\left[\pi, \frac{3 \pi}{2}\right)\) and that the range of arccosecant is \(\left(0, \frac{\pi}{2}\right] \cup\left(\pi, \frac{3 \pi}{2}\right]\) when finding the exact value. \(\operatorname{arccsc}\left(\csc \left(\frac{9 \pi}{8}\right)\right)\)

In Exercises \(99-107\), express the domain of the function using the extended interval notation. (See page 756 in Section 10.3 .1 for details.) $$ f(x)=\ln (|\cos (x)|) $$

Find the domain of the given function. Write your answers in interval notation. $$ f(x)=\operatorname{arccsc}\left(e^{2 x}\right) $$
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