Relation Between Degree and Radian Measures

Introduction

Angles are often measured in both the degree and radian systems. The degree system is based on a circle with 360°, while the radian system is based on a circle with a circumference equal to the product of its diameter and π. To convert between these two measurement systems, you use the following equation:

angle = degrees × (π/180)

The Degree System

The degree system is the most common way of measuring angles. It is based on a circle of 360°. Consequently, a degree is defined as 1/360 of a turnaround. A full 360° turn is required to form a complete circle. Given that this is often too cumbersome to use practically in all circumstances, the degree system is based on a circle with 360°/1 = 1 revolution.

When measuring an angle, the size of that angle is compared against that of a full circle. A degree is defined as 1/360 of a full circle. This works out to be about 57.3°, since 360° = 1 revolution, and there are 360 degrees in a revolution.

The Radian System

The radian system is based on a circle with a circumference equal to the product of its diameter and π. A radian is defined as the angle formed by an arc with a length equal to the radius of the circle it forms on, divided by its circumference.

(1/2π) radians = 360°

Since a revolution is 2π radians, it can be assumed that there are 2π radians in a full circle. This is why the conversion equation between degrees and radians is Degrees = Radians * 180/π.

Relation Between Degree And Radian Measures 

The following equation is used in conversions between degrees and radians:

radians = degrees × (π/180)

Since there are 180° in a full circle, or equivalently 360° in a complete turn, this conversion equation is equivalent to:

degress = (2π/φradians) 180°

The value φ, about 3.14159, is known as the circle constant or the pi constant. The angle corresponding to this conversion equation is called the radian measure of the degree angle.

Comparison between Degree and Radian Measures

A radian is a universal unit of measure. It can measure any angle, whether it is in the degree or the radian system. As a result, two angles with different units of measure are often compared against each other.

For example, if one angle is measured in degrees and another is measured in radians, the two angles may be compared by converting them to a standard unit of measure.

It is generally easier to work with angles in the radian system. However, it is important to know how to use the conversion equation between the two systems so that angles can be converted between the two systems when necessary. When working with angles in the degree system, it is essential to remember that there are 360° in a full circle.

Working with these systems of measuring angles allows you to easily divide angles by 360 to get a percentage of the circle that the angle occupies. 

For example, an angle measuring 45° is about 1/4th of a circle, or 90°. 

Conversely, when working with angles in the radian system, it is essential to remember that there are 2π radians in a full circle.

This makes it easy to divide angles by 2π to get the percentage of the circle that the angle occupies. 

For example, an angle measuring π/6 radians is about 1/12th of a circle, or 30°.

Derivation of Degree to Radian conversion

The conversion equation between degrees and radians can be derived using basic trigonometric identities. In particular, we will use the following identity:

sin(θ) = θ/r

where θ is the angle in radians, r is the circle’s radius, and sin(θ) is the sine of the angle θ.

First, one revolution = 360° must be proved using this equation. This is done by setting up a proportion, which says that:

sin(1) * r = θ Substituting this into the identity given above gives:

θ = sin(1)/r Thus, it is proved that 1 revolution = 360°.

Understanding how this proof works is essential in situations that do not involve angles. The proof works because the sine function is an increasing function on its domain (-1 to +1). This means that there are infinite solutions for θ, which is needed to prove that 1 revolution = 360°. In particular, sin(θ) will be equal to θ when θ is in the first quadrant (0° < θ < 90°), and this is the angle of interest when measuring angles in degrees.

The identity sin(θ) = θ/r may now be used to derive the conversion equation between degrees and radians. In particular, if φ = r * 2π, it implies that:

sin(θ) = θ/φ or sin(θ) = 1/φ 

This identity will allow us to find a relationship between degrees and radians.

Dividing both sides of this equation by sin(θ), we get:

φ = θ/sin(θ) or φ = 1/(cos(θ)) 

This is the conversion equation between degrees and radians. This derivation can also be found in any trigonometry textbook.

Radian and Degree as Units of Measurement for an Angle

When using radians as a unit of measure, it is often more convenient to use them with trigonometric functions. This is because the sine and cosine of an angle in radians are equal to:

sin(θ) = θ

cos(θ) = 1/θ

This is much more straightforward than the equation given in the previous section.

Relation Between Degree and Radian Measures

Relationship between degrees and radians:

sin(θ) = θ/r

φ = 1/(cos(θ))

for 0 ≤ θ < 2π; equivalently, if we set π = 180°:

sin(θ) = θ/180

This enables a conversion between degrees and radians. In particular:

θ = sin-1((180/π) * y), where y is a distance measured in radians.

On the other hand, if we want to convert an angle from degrees to radians, we can use the following formula:

θ = 180sin(180/π) * y

where y is a distance measured in degrees.

Conclusion

Knowing the conversion equation between degrees and radians makes the conversion of angles between the two systems easy to do when necessary. It is also essential to know which unit of measurement any given angle is expressed in so that the appropriate trigonometric function may be used while working with it. It is often more convenient to work with angles in radians due to the simplicity of using sine and cosine functions. However, if the problem in question is using degrees as the integral unit of measurement, the conversion equation may be used to easily move between the two systems.