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Chapter 6: Problem 19

Find the degree measure of the angle with the given radian measure. $$ 3 $$

Short Answer

Expert verified
The degree measure is \(\frac{540}{\pi}\).

Step by step solution

01

Understand the Relationship between Radians and Degrees

The relationship between radians and degrees is defined through the fact that a full circle is 360 degrees or \(2\pi\) radians. This means that \(1\text{ radian} = \frac{180}{\pi}\text{ degrees}\). We will use this conversion factor to change radians into degrees.
02

Convert Radian Measure to Degrees

Given the radian measure \(3\), we will convert this into degrees using the conversion formula:\[\text{Degrees} = 3 \times \frac{180}{\pi}\]Simplifying this expression will give us the degree measure.
03

Simplify the Expression

Let's simplify the expression \(3 \times \frac{180}{\pi}\):1. Calculate the multiplication: \(3 \times 180 = 540\).2. Divide the result by \(\pi\): \(\frac{540}{\pi}\). This gives us the degree measure of the angle.

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Key Concepts

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

Angle Measurement
Angles help us describe the size or rotation of a shape. They are especially useful in geometry, physics, and engineering. When measuring angles, we usually use either degrees or radians as units. Each type of measurement has its own advantages depending on the context.
  • Degrees use a base of 360 to represent a full circle. This division makes it easy for people to understand and visualize angles.
  • Radians, on the other hand, are often more natural in mathematics because they align with the properties of circles in trigonometric and calculus calculations.
Knowing how to convert between degrees and radians allows us to work easily across different fields and applications.
Degrees and Radians
Degrees and radians are two ways to express the measure of an angle. Understanding both can enhance our ability to solve problems in mathematics.
  • Degrees: A full circle is divided into 360 equal parts, each part being one degree. This system is intuitive because it divides the circle into segments which are multiples of 10, a format familiar in daily life.
  • Radians: A radian measures the angle created when the arc length is equal to the radius of the circle. A full circle is therefore \(2\pi\) radians. This unit is crucial for complex calculations in mathematics because it incorporates the circle's geometry directly.
Both systems are important, and depending on the context, one might be more convenient than the other.
Conversion Factor for Angles
Converting between degrees and radians is straightforward if you remember the conversion factor. A circle with 360 degrees corresponds to \(2\pi\) radians. Therefore, we can determine that:
  • \(1\text{ radian} = \frac{180}{\pi}\text{ degrees}\).
  • \(1\text{ degree} = \frac{\pi}{180}\text{ radians}\).
To convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\). Similarly, convert from degrees to radians by multiplying the degree measure by \(\frac{\pi}{180}\).
Understanding and remembering this conversion factor allows us to easily transition between the two measurement systems, making calculations and comprehension in mathematics much smoother.

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Most popular questions from this chapter

Beehives In a beehive each cell is a regular hexagonal prism, as shown in the figure. The amount of wax \(W\) in the cell depends on the apex angle \(\theta\) and is given by $$ W=3.02-0.38 \cot \theta+0.65 \csc \theta $$ Bees instinctively choose \(\theta\) so as to use the least amount of wax possible. (a) Use a graphing device to graph Was a function of \(\theta\) for \(0 < \theta < \pi\) (b) For what value of \(\theta\) does \(W\) have its minimum value? [Note: Biologists have discovered that bees rarely deviate from this value by more than a degree or two. \(]\)

Sledding The time in seconds that it takes for a sled to slide down a hillside inclined at an angle \(\theta\) is $$ t=\sqrt{\frac{d}{16 \sin \theta}} $$ where \(d\) is the length of the slope in feet. Find the time it takes to slide down a \(2000-\mathrm{ft}\) slope inclined at \(30^{\circ} .\)

An isosceles triangle has an area of \(24 \mathrm{cm}^{2},\) and the angle between the two equal sides is \(5 \pi / 6 .\) What is the length of the two equal sides?

\(19-28\) . 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 area of a triangle with sides of length 10 and 22 and included angle \(10^{\circ} .\)
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