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Chapter 13: Problem 25

Rewrite each degree measure in radians and each radian measure in degrees. \(\frac{11 \pi}{4}\)

Short Answer

Expert verified
\(\frac{11\pi}{4}\) in degrees is \(495^\circ\).

Step by step solution

01

Convert Radian to Degrees

To convert from radians to degrees, use the formula \( heta( ext{degrees}) = heta( ext{radians}) \times \frac{180}{ ext{蟺}} \). This formula arises because \( ext{蟺} ext{ radians} = 180^ ext{o} \).
02

Substitute the Given Radian

Substitute \( \frac{11 ext{蟺}}{4} \) into the conversion formula to find its degree measure: \( \frac{11 ext{蟺}}{4} \times \frac{180}{ ext{蟺}} \).
03

Simplify the Expression

Notice that \( \text{蟺} \) in the numerator and denominator cancels out, simplifying the expression to \( \frac{11}{4} \times 180 \).
04

Compute the Multiplication

Calculate \( \frac{11}{4} \times 180 \), which results in \( \frac{1980}{4} \).
05

Perform Division to Get the Answer

Divide 1980 by 4 to get the degree measure: \( 495^ ext{o} \).

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Key Concepts

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

Radian Measure
A radian is a unit of angular measure used in many areas of mathematics. It is based on the radius of a circle. Imagine wrapping a length of the circle's radius around its edge; this length would define an angle of one radian at the circle's center. This makes radians a natural way to express angles when dealing with circular or periodic objects.

Unlike degrees, which divide a circle into 360 equal parts, radians divide the circumference of a circle in relation to its radius. Typically, a full circle covers an angle of \(2\pi\) radians. Thus, radians provide a straightforward way to relate angles to physical dimensions.
Degree Measure
Degrees offer a familiar way to measure angles in everyday settings. A degree is defined as \(\frac{1}{360}\) of a complete rotation around a circle's center.

Degrees are easy to visualize and thus frequently employed in fields like geography and engineering. However, when dealing with functions of periodic nature or differential equations, radians become more convenient.
  • One full circle: 360 degrees
  • One right angle: 90 degrees
  • Degrees breakdown allows easy measurement and communication
Mathematical Formula
Transforming between radians and degrees is essential for problems involving angular measurements. This is where a mathematical formula becomes invaluable:

Conversion from radians to degrees utilizes the relationship: \( \theta(\text{degrees}) = \theta(\text{radians}) \times \frac{180}{\pi} \).
Here's the reasoning: Since \(\pi\) radians equal 180 degrees, multiplying a radian measure by \(\frac{180}{\pi}\) converts it to degrees.
  • It translates radians (a more abstract concept) into degrees (a more tangible one).
  • Provides an effortless way to switch between the two measurement units.
Simplification Steps
Converting \( \frac{11\pi}{4} \) radians to degrees involves a series of simplification steps:

  • **Multiply by the conversion factor.** Substitute the given radian measure into the conversion formula, multiplying by \(\frac{180}{\pi}\).
  • **Cancel out values.** Notice that \(\pi\) itself cancels out, leaving you with a simpler expression.
  • **Perform arithmetic calculations.** You are left with the operation \(\frac{11}{4} \times 180\).
  • **Finalize with division.** After performing the multiplication, divide the result \(1980\) by \(4\) to obtain the final answer: \(495^\circ\).
These steps allow you to transition from a mathematical expression in radians to a more practical degree measure. In many educational and practical scenarios, such clear understanding of these basic transformations is crucial.

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Most popular questions from this chapter

FORESTRY Two forest rangers, 12 miles from each other on a straight service road, both sight an illegal bonfire away from the road. Using their radios to communicate with each other, they determine that the fire is between them. The first ranger's line of sight to the fire makes an angle of 38 with the road the second ranger's line of sight to the fire makes a \(63^{\circ}\) angle with the road. How far is the fire from each ranger?

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\sin \left(2 \sin ^{-1} \frac{1}{2}\right)\)

Solve each equation. Round to the nearest tenth. $$ 13^{2}=8^{2}+6^{2}-2(8)(6) \cos A^{\circ} $$

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\tan (\operatorname{Arctan} 3)\)

Find each value. Write angle measures in radians. Round to the nearest hundredth. $$ \cos ^{-1}\left(-\frac{1}{2}\right) $$
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