Question

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

Short answer

720 degrees

Step-by-step solution

- Understand the Radian to Degree Conversion Formula

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

- Substitute the Radian Value

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

- Simplify the Expression

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

- Multiply to Get the Degree Measure

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

Key concepts

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.