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Chapter 7: Problem 40

Convert each angle in radians to degrees. \(4 \pi\)

Short Answer

Expert verified
720 degrees

Step by step solution

01

- Understand the Radian to Degree Conversion Formula

To convert an angle in radians to degrees, use the conversion factor \ \( 1 \text{ radian} = \frac{180}{\text{蟺}} \text{ degrees} \ \). This is based on the fact that \ \( \text{蟺 radians} \) is equal to \ \( 180 \text{ degrees} \). Hence, multiply the radian measure by \ \( \frac{180}{\text{蟺}} \).
02

- Substitute the Radian Value

Substitute the given radian value \ \( 4 \text{蟺} \) into the conversion formula. This gives: \ \[ 4 \text{蟺} \times \frac{180}{\text{蟺}} \].
03

- Simplify the Expression

Simplify the expression by canceling out \ \( \text{蟺} \) in the numerator and the denominator, which results in: \ \[ 4 \times 180 \].
04

- Multiply to Get the Degree Measure

Multiply the remaining numbers: \ \[ 4 \times 180 = 720 \]. Hence, the angle in degrees is \ 720 \text{ degrees}.

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Key Concepts

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

Understanding Radians
In trigonometry, radians are a unit of angular measure. One radian is defined as the angle formed when the length of the arc of a circle is equal to the radius of the circle. Radians provide a natural way to express angles, especially when dealing with calculus and other advanced mathematics.
Understanding Degrees
Degrees are another unit of angular measure. Unlike radians, degrees divide a circle into 360 equal parts. This ancient Babylonian system is pervasive in daily life and is often used in navigation, engineering, and many other fields.
Angle Conversion
Converting between radians and degrees is crucial in trigonometry. The conversion factor is derived from the fact that \( \pi \text{ radians} \) equals \( 180 \text{ degrees} \). To convert an angle from radians to degrees, multiply the radian value by \( \frac{180}{\pi} \).
For example, to convert \( 4\pi \) radians to degrees: \[ 4\pi \times \frac{180}{\pi} = 720 \text{ degrees} \].
Basic Trigonometry Concepts
Trigonometry is the study of the relationships between the angles and sides of triangles. It includes concepts such as sine, cosine, and tangent, which are functions that relate the angles of a triangle to the lengths of its sides. Conversion between radians and degrees is one of the foundational skills in trigonometry, enabling the use of trigonometric functions in a variety of calculations and applications.

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Most popular questions from this chapter

Convert each angle in degrees to radians. Express your answer in decimal form, rounded to two decimal places. \(73^{\circ}\)

Use a coterminal angle to find the exact value of each expression. Do not use a calculator. $$ \cos 420^{\circ} $$

\(f(x)=\sin x, g(x)=\cos x, h(x)=2 x,\) and \(p(x)=\frac{x}{2} .\) Find the value of each of the following: $$ (f \circ h)\left(\frac{\pi}{6}\right) $$

Geometry The hypotenuse of a right triangle has a length of 10 centimeters. If one angle is \(40^{\circ},\) find the length of each leg.

\(f(x)=\sin x, g(x)=\cos x, h(x)=2 x,\) and \(p(x)=\frac{x}{2} .\) Find the value of each of the following: $$ (h \circ f)\left(\frac{\pi}{6}\right) $$
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