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Chapter 6: Problem 21

Find the degree measure of the angle with the given radian measure. $$\frac{\pi}{10}$$

Short Answer

Expert verified
18 degrees.

Step by step solution

01

Convert Radians to Degrees

To convert the given radian measure to degrees, use the conversion formula: degrees = radians \(\times \frac{180}{\pi}\).
02

Apply the Conversion Formula

Insert the given radian measure \(\frac{\pi}{10}\) into the conversion formula: Degrees = \(\frac{\pi}{10} \times \frac{180}{\pi}\).
03

Simplify the Expression

Cancel the \(\pi\) from the numerator and denominator: \(\frac{180}{10}\).
04

Calculate the Degree Measure

Perform the division: \(\frac{180}{10} = 18\). Thus, the degree measure is 18 degrees.

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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 way of describing angles using the radius of a circle. It's a unit that tells us how much an angle opens, based on the proportion of the circle's circumference it sweeps through.

Imagine slicing a cake with a point at the center. If you want a slice that comprises exactly one "pi" (or 蟺) part of the full circle, you're thinking in radians. In simpler terms:
  • 1 radian is the angle made when the arc length is equal to the radius of the circle.
  • The entire circle is 2蟺 radians, just as it is 360 degrees.
  • Thus, when dealing with angles in terms of the circle, radians are extremely handy.
Radian measure simplifies many equations in mathematics, especially in calculus and trigonometry.
Degree Measure
Degree measure is a more familiar way to describe angles, often used in everyday contexts like maps and geometry. A full circle is divided into 360 degrees, and this number helps us to visualize and communicate angle sizes more intuitively.

For example, when you hear that something is at a 90-degree angle, you can easily picture a right angle. Some key points about degree measure include:
  • Every degree is one 360th of a full circle.
  • It's often more practical for measurements in everyday life due to its simplicity.
  • Degree measures make it easier to convey angle sizes quickly without needing mathematical calculations.
Overall, while degrees might not always be as mathematically powerful as radians in specific contexts, they're crucial for traditional and everyday uses.
Conversion Formula
Converting between radians and degrees is essential, as different types of problems might require different angle measurements. Fortunately, this conversion can be done using a straightforward formula.

To convert from radians to degrees, the formula is:
  • degrees = radians 脳 \(\frac{180}{\pi}\)
This formula works because 180 degrees is equivalent to \(\pi\) radians, which allows us to scale one unit to another. For instance, to find how many degrees are in \(\frac{\pi}{10}\) radians, you simply substitute \(\frac{\pi}{10}\) into the formula:
  • Multiply: \(\frac{\pi}{10} \times \frac{180}{\pi} = \frac{180}{10} \)
  • Calculate: \(\frac{180}{10} = 18\)
As seen, given these conversions, one radian equals approximately 57.2958 degrees, which can assist you in double-checking your work.

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Most popular questions from this chapter

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. $$\begin{array}{|l|l|l|l|l|} \hline \theta & 20^{\circ} & 60^{\circ} & 80^{\circ} & 85^{\circ} \\ \hline h & & & \\ \hline \end{array}$$

Use the Law of sines to solve for all possible triangles that satisfy the given conditions. $$a=28, \quad b=15, \quad \angle A=110^{\circ}$$

Find the exact value of the trigonometric function. $$\cos 570^{\circ}$$

Speed of a Car The wheels of a car have radius 11 in. and are rotating at 600 rpm. Find the speed of the car in \(\mathrm{mi} / \mathrm{h}\).

\(A 600\) -ft guy wire is attached to the top of a communications tower. If the wire makes an angle of \(65^{\circ}\) with the ground, how tall is the communications tower?
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