How Degree and Radian Measures are Related to Each Other

The degree and radian measure of an angle are related to one another. The degree measure specifies the size of an angle concerning a full circle, whereas the radian measure specifies the size of an angle concerning itself. Although larger angles may be specified with a smaller degree measurement, it does not imply that it will be easier to draw or construct. Further, in this article, we are also going to discuss solved examples of degree and radian. 

About Degree and Radian

A radian is a unit of measurement which is commonly used in trigonometry. A radian is equal to one-half of the length of a circle’s arc, or the distance between two points on a circle, where one point is at the centre and the other point is on the circumference. 

A degree is another unit of measurement commonly used in trigonometry, and it is equal to one-hundredth of a circle’s arc or three hundred sixty-five degrees divided by two. Degrees can also be defined as angles from zero to ninety degrees, with zero being equivalent to facing directly north.

Metres, feet, inches and other traditional units of measurement are all based on these two units.

The symbol for radians is “rad”, and its abbreviation is “rad”. The symbol for degrees is “deg”, and its abbreviation is “°”.

Radian Measure Formula:

In any circle of radius r, the ratio of the arc length ℓ to the circumference is equivalent to the ratio of the angle θ subtended by the arc at the midpoint and the angle in one rotation. Therefore, （tanθ=r△y/x△z）

Degree Calculation Formula: 

To calculate degrees, you first need to find the angle’s measure. The measure is the angle’s size, which is its degree measurement. To determine this, use the following formula: 

Radians × 180/π = Degrees

Solved Examples of Degree and Radian

Following are some solved examples of degree and radian for better understanding: 

Example 1: Convert 135° into a radian measure

Solution: Given = Angle of 135 degrees135°× π/180°=3π/4 radians≈2.35 radians

Example 2: Convert 210 degrees to radians.

Solution: Given = Angle of 210 degrees Angle in radian = Angle in degree x (π/180)= 210 x (π/180)= 7π/6≈3.67Hence, 210 degrees is equal to 7π/6 (3.67) in radian.

Example 3: Convert 200 degrees into radians.

Solution: By the formula, we know;

Angle in radians = Angle in degree × π/180

Thus,

200 degrees in radians = 200 × π/180 = 10π/9 = 3.491 Rad

Example 4: Convert 450 degrees into radians.

Solution: By the formula, we know;

Angle in radians = Angle in degree × π/180

Thus,

450 degrees in radians = 450 × π/180 = 7.854 Rad

Therefore, with the above-solved examples of degree and radian, we came up with two important formulas: 

• Radians × 180/π = Degrees
• ℓ2πr=θ2π⟹ ℓ=rθ

What is the relation between degree and radian?

 The answer to the question of what is the relationship between radian and degree is that one radian is equal to 180/π degrees. When a radius is rotated through one radian, the arc length is equal to the radius times the cosine of the angle, which can be simplified as follows:

The cosine of an angle equals its adjacent side over the hypotenuse. If you know the radius and angle, you can find the length of the arc by multiplying them together. The angle in radians is equal to its degrees times π. So you can say that one degree is equal to π/180 radians, which equals 0.0174533… radians. By dividing 180 into π, you get 180/π, which simplifies to 60°. This means that each degree has 60/π radians or 0.0174533…radians. This makes it easy to convert between degrees and radians, but they are not interchangeable because they are different units of measure — you cannot just replace a number with “degrees” or “radians” to convert it from one unit of measure to another without knowing what the correct answer should be.

Important Uses of Degree and Radian

We use radian measure all the time. There are many important uses of degree and radian. Radian measure is a unit used to measure angles. As you know, there are 360° in a circle. If you divide that by 2π, you get 57.3° per radian. If you think of this literally as measuring an angle, it would mean that one radian is equal to the angle formed by a circle at its centre. Degree and radian are both useful ways to measure angles; they are just different units of measurement.

We use degree measures when we write pedagogical materials for students who are learning the basics of trigonometry. This is because students in these classes tend to be more familiar with degrees than with radians, and degree measure takes up less space in equations and on the page than does radian measure (in contrast, we often use radians for research papers because it takes up less space than degrees).

In general, whether one uses degree or radian measure depends on personal preference and context. 

Conclusion

Degrees and radians are both ways to measure angles. We use degrees because they’re convenient to work with and most people are familiar with them. Radians are based on the amount of distance travelled around the circumference of a circle, which is why they’re used in describing the sine, cosine, and tangent functions. In a right triangle, where one side is touching the circle’s centre and another side wraps around the edge of the circle, the hypotenuse equals 1, and the ratio of this right triangle’s sides equals the trigonometric function of its angle.