Radians vs Degrees: Why Calculus Uses Radians

In summary, radians are used in calculus instead of degrees because they have nice properties for derivatives and allow functions to be defined for a wider range of numbers. They are also defined as a distance rather than an angle, unlike degrees, which are often associated with angles. This makes them a more suitable unit for mathematical operations.
  • #1
chandran
139
1
why radians are used in calculus instead of degrees?
 
  • Like
Likes seekerstein
Physics news on Phys.org
  • #2
Radians have the very nice property that [tex]lim_{x->0}\frac{sin x}{x}[/tex]= 1[/tex] and [tex]lim_{x->0}{1- cos x}{x}= 1[/tex] when x is in radians. As a result the derivative of sin x is cos x and the derivative of cos x is -sin x as long as x is in radians. That's not true if x is measured in degrees. If we insist upon using degrees the corresponding derivatives would be multiplied by [tex]\frac{180}{\pi}[/tex].


That's the easy answer. A little deeper- we don't define sine and cosine, in "higher" mathematics in terms of right triangles at all: in a right triangle would have to be between 0 and 90 degrees and we want functions to be defined as many numbers as possible. One definition widely used is this: We are given an xy-coordinate system and the unit circle (the graph of the relation x2+ y2= 1). To find sin t and cos t (for t non-negative), measure around the circumference of the circle, counter clockwise, a distance t (if t< 0, measure clockwise a distance -t). The point at which you end has coordinates, by definition, cos t and sin t. ("by definition"- in other words, whatever the coordinates are, that is how we define cos t, sin t.)

Notice that the variable t in that definition is not measured in degrees OR radians! It is a distance, not an angle. Unfortunately, calculators are designed by engineers, not mathematicians and engineers tend to think of sine and cosine in terms of angles ("phase angles" in electromagnatism have nothing to do with angles!). "Radians" are defined so that the radian measure of an angle is the same as the length of the arc on a unit circle.
 
  • #3


Radians and degrees are two units of measurement used to measure angles. Degrees are more commonly used in everyday life, while radians are primarily used in mathematics, particularly in calculus.

One of the main reasons why radians are preferred in calculus is because they are a more natural unit of measurement for angles in mathematics. Radians are defined as the ratio of the arc length to the radius of a circle, while degrees are based on dividing a circle into 360 equal parts. This means that radians are a more fundamental and intuitive unit of measurement for angles, as they are directly related to the geometry of a circle.

Furthermore, radians are also more convenient to use in calculus because they simplify trigonometric functions. In calculus, trigonometric functions such as sine and cosine are used extensively to model and solve problems. When angles are measured in radians, these functions can be expressed in terms of simple fractions and their derivatives can be easily calculated. This makes working with trigonometric functions much easier and more efficient in calculus.

Moreover, radians also have a unique property that makes them more suitable for calculus – the arc length of a circle is equal to the angle in radians. This property is known as the "unit circle property" and is crucial for understanding and solving problems in calculus involving circular motion and other related concepts.

In conclusion, radians are used in calculus instead of degrees because they are a more natural and fundamental unit of measurement for angles in mathematics, simplify trigonometric functions, and have a unique property that is essential in calculus. While degrees are more commonly used in everyday life, radians are the preferred unit of measurement in the world of mathematics and specifically in calculus.
 

Related to Radians vs Degrees: Why Calculus Uses Radians

1. What is the difference between radians and degrees?

Radians and degrees are units used to measure angles. Degrees are based on a circle divided into 360 equal parts, while radians are based on the length of the arc of a circle equal to the radius of the circle.

2. Why does calculus use radians instead of degrees?

Calculus uses radians because it simplifies mathematical calculations involving angles and circular motion. Radians are a more natural unit for calculus because they directly relate the angle to the length of the arc, making it easier to work with in equations.

3. Can radians and degrees be converted to each other?

Yes, radians and degrees can be converted to each other using conversion formulas. To convert from degrees to radians, multiply the angle by π/180. To convert from radians to degrees, multiply the angle by 180/π.

4. Are radians used in any other fields besides calculus?

Yes, radians are commonly used in physics, engineering, and other fields that involve circular motion, such as astronomy and navigation. They are also used in trigonometry and geometry.

5. Is there an advantage to using radians over degrees?

Yes, there are several advantages to using radians over degrees. Radians are a more natural and intuitive unit for measuring angles in calculus and other mathematical calculations. They also make it easier to work with trigonometric functions and simplify equations. Additionally, many calculus and physics concepts, such as derivatives and integrals, are defined using radians.

Similar threads

  • Calculus
Replies
11
Views
10K
Replies
8
Views
2K
Replies
11
Views
2K
  • General Math
Replies
16
Views
1K
  • Calculus
Replies
18
Views
4K
Replies
15
Views
4K
  • General Math
Replies
9
Views
2K
  • Introductory Physics Homework Help
Replies
14
Views
2K
Replies
4
Views
4K
Back
Top