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Chapter 6: Problem 14

\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ \frac{11 \pi}{3} $$

Short Answer

Expert verified
The angle measures 660 degrees.

Step by step solution

01

Understand the Problem

We are given an angle in radians, \( \frac{11\pi}{3} \), and need to convert it to degrees. Recall that \( 180^\circ = \pi \) radians.
02

Use the Radian to Degree Conversion Formula

The formula to convert radians to degrees is \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \). Substituting \( \frac{11\pi}{3} \) for Radians gives us: \( \frac{11\pi}{3} \times \frac{180}{\pi} \).
03

Simplify the Expression

Start by cancelling \( \pi \) from the numerator and the denominator: \( \frac{11}{3} \times 180 \).
04

Calculate the Degrees

Now, compute the multiplication: \( \frac{11}{3} \times 180 = 11 \times 60 = 660 \).
05

Conclusion

The angle with radian measure \( \frac{11\pi}{3} \) converts to \( 660^\circ \).

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Key Concepts

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

Radian Measure
Radian measure is a unit of angular measure used in mathematics. It is based on the radius of a circle. When you think of radian measure, you can visualize it as the angle created by taking the radius of a circle and wrapping it along the circle's circumference. This method of measuring angles is quite natural for calculations in trigonometry, calculus, and many fields of physics.
  • One complete revolution around a circle corresponds to an angle of 2\(\pi\) radians.
  • A straight line (or half a circle) corresponds to an angle of \(\pi\) radians.
  • Smaller angles can be expressed as fractions or multiples of \(\pi\).
Radian measure is dimensionless, meaning it doesn't rely on external units like degrees. This often simplifies mathematical calculations. In our exercise, the angle \( \frac{11\pi}{3} \) is given in radians, which can be broken down into more intuitive rotations around the circle.
Degree Measure
Degree measure is another way to express angles, commonly used in navigation, engineering, and everyday settings. Degrees break a circle into 360 equal parts. This format can be easier to comprehend in non-mathematical contexts.
  • One complete circle is 360 degrees.
  • Half a circle, which is a straight line, is 180 degrees.
  • Quarter of the circle, a right angle, is 90 degrees.
Degrees are often more intuitive when dealing with everyday problems or when visualizing angles in a typical setting, such as the corners of a square or the direction of a compass. Understanding the degree measure allows you to easily relate mathematical concepts to real-world scenarios.
Angle Conversion Formula
The angle conversion formula is a bridge between radian and degree measures. It allows one to translate angles from one unit to the other efficiently. The formula is expressed as:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]Here's how it works:
  • To convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).
  • This factor comes from the fact that \(\pi\) radians equal 180 degrees.
In the exercise, converting \( \frac{11\pi}{3} \) radians into degrees involved substituting into the formula:\[\frac{11\pi}{3} \times \frac{180}{\pi} = 660\degree\]We first cancel \(\pi\) from both the numerator and the denominator, then multiply \(\frac{11}{3}\) by 180 to get the final degree measurement. This simple and systematic approach allows you to switch between angle measures effortlessly.

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Most popular questions from this chapter

Turning a Corner A steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. (a) Show that the length of the pipe in the figure is modeled by the function $$L(\theta)=9 \csc \theta+6 \sec \theta$$ (b) Graph the function \(L\) for \(0<\theta<\pi / 2\) (c) Find the minimum value of the function \(L\) (d) Explain why the value of \(L\) you found in part (c) is the length of the longest pipe that can be carried around the corner.

A man is lying on the beach, flying a kite. He holds the end of the kite string at ground level, and estimates the angle of elevation of the kite to be \(50^{\circ} .\) If the string is 450 ft long, how high is the kite above the ground?

Rain Gutter A rain gutter is to be constructed from a metal sheet of width 30 \(\mathrm{cm}\) by bending up one-third of the sheet on each side through an angle \(\theta\) . (a) Show that the cross-sectional area of the gutter is modeled by the function $$A(\theta)=100 \sin \theta+100 \sin \theta \cos \theta$$ (b) Graph the function \(A\) for \(0 \leq \theta \leq \pi / 2\) (c) For what angle \(\theta\) is the largest cross-sectional area achieved?

Two boats leave the same port at the same time. One travels at a speed of 30 mi/h in the direction N \(50^{\circ} \mathrm{E}\) and the other travels at a speed of 26 \(\mathrm{mi} / \mathrm{h}\) in a direction \(\mathrm{S} 70^{\circ} \mathrm{E}\) (see the figure). How far apart are the two boats after one hour?

Find the area of the triangle whose sides have the given lengths. \(a=7, \quad b=8, \quad c=9\)
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