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Chapter 6: Problem 8

Find the radian measure of the angle with the given degree measure. $$3960^{\circ}$$

Short Answer

Expert verified
The radian measure is \( 22\pi \).

Step by step solution

01

Understand Degree to Radian Conversion

Remember that to convert degrees to radians, we use the formula: \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \). This means we multiply the degree measure by \( \frac{\pi}{180} \) to obtain the radian measure.
02

Set Up the Conversion Formula

Plug in \( 3960^{\circ} \) into the conversion formula from Step 1: \[ 3960 \times \frac{\pi}{180} \].
03

Simplify the Conversion Expression

Evaluate the expression by simplifying the fraction first. Divide 3960 by 180 to get: \[ 3960 \div 180 = 22 \]. Now the expression becomes \( 22\pi \).

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Key Concepts

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

Radians
Radians are a unit of angular measurement used primarily in mathematics. They're part of the International System of Units (SI) and provide a natural way to describe angles in terms of the radius of a circle.
One radian is defined as the angle created when we take the radius of a circle and wrap it along the circle's edge. This makes it easier to relate angles directly to the radius length.
  • When studying trigonometry or calculus, you'll often see radians used instead of degrees.
  • Radians offer a clean, straightforward approach to calculations because of their direct connection to circular geometry.
Understanding the radian measure allows you to grasp circular motion concepts more naturally, which plays a key role in advanced mathematics and physics.
Degrees
Degrees are the more familiar measure of angles, particularly in everyday contexts and basic geometry. One full rotation around a circle is divided into 360 degrees, making each degree quite a small angle.
  • The concept of degrees is ancient and likely originated from the Babylonians, who used a base-60 number system.
  • Degrees are very intuitive, which makes them great for simple angle measurement and navigation.
While degrees are more common in many fields, converting them to radians is crucial in higher-level mathematics. This conversion facilitates mathematical calculations and understanding, especially in areas involving periodic and circular functions.
Angle Measurement
Angle measurement is a core concept in both mathematics and applied sciences. It involves determining the size of an angle, which describes the opening between two line segments or rays.
  • Angle measurements can be expressed in different units, such as degrees, radians, or even gradians.
  • Choosing the right unit often depends on the context of the problem or task at hand.
In mathematics and physics, radians are often preferred due to their natural relationship with the properties of a circle. Meanwhile, degrees offer a more intuitive approach in everyday applications, such as in construction or navigation.
Understanding how to switch between these units, like converting degrees to radians, is vital. It ensures precision and suitability in calculations across different scientific and engineering fields.

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Most popular questions from this chapter

Use the Law of sines to solve for all possible triangles that satisfy the given conditions. $$a=75, \quad b=100, \quad \angle A=30^{\circ}$$

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\tan \theta=-4, \quad \sin \theta>0$$

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?

Height of a Rocket \(A\) rocket fired straight up is tracked by an observer on the ground a mile away. (a) Show that when the angle of elevation is \(\theta,\) the height of the rocket in feet is \(h=5280 \tan \theta\). (b) Complete the table to find the height of the rocket at the given angles of elevation. $$\begin{array}{|l|l|l|l|l|} \hline \theta & 20^{\circ} & 60^{\circ} & 80^{\circ} & 85^{\circ} \\ \hline h & & & \\ \hline \end{array}$$

A parallelogram has sides of lengths 3 and 5 and one angle is \(50^{\circ} .\) Find the lengths of the diagonals.
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