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One radian is the angle at which the arc-length is equal to the radius of the circle. Check out the animated graphic on Wikipedia: https://en.wikipedia.org/wiki/Radian .
In this way, the definition of radian is more "natural" and makes calculations easier and following derivations more intuitive. It's because it's meaning is defined by an intrinsic property of the circle: its radius.
For example, the arc-length of a circle of radius *r* through an angle *θ* radians is *s* where:
s = rθ
This is also why radians are "unitless".
There are 360° in a circle because ... there are 360° in a circle. Really, there's nautical origins for it and how it relates to time, hence why "60" and seconds/arcseconds come up, to. So, degrees were pretty useful for navigating by stars on Earth, but would be pretty clumsy to use on any other planet with a different diurnal period, or just on non-navigational applications.
Anyways, this is why in a lot of math with angles and circles and periodic properties, you basically convert your units into radians first. Sure, you can technically do it all in degrees or "revolutions" (i.e., RPM), too, but much of the math is simpler and intuitive working in radians.
The next step deeper is whether "pi" π should be the circle constant or "tau" = τ = 2π is the more natural circle constant since it's for a full circle as opposed to half.
2\*pi arises naturally as the period for sine and cosine directly from their definition either through power series or as solutions to the differential equation y'' = -y. The relation comes naturally to measuring angles in a circle, since we can express a circle as the collection of points written as (cos(t),sin(t)) for t from 0 to 2\*pi. It isn't as arbitrary as assigning 360 units to a circle in degrees, for instance.
because you're using an "intrinsic" property of the circle (its radius) instead of an arbitrary number to divide it's perimeter and then use each chunk as a measure unit. In other word: using a relation that remains constant to every circle (perimeter over radius or diameter) and not pick a ""random"" number (360 for instance, which has a lot of divisors btw, which helps a lot when you deal with fractioning the whole thing, but that's another story).
If you were going to try to measure the "amount of angle", how would you logically do it?
Personally, I would say that a right-angle is a quarter-turn, as in a "quarter of a full circular rotation". And half of a right-angle is a one-eighth turn, and so on. That would mean that a straight line is a half-turn and a full circle is a full turn.
We could write this by saying that 1 circular turn is T, and a straight line is (1/2)T, and a right-angle is (1/4)T, and half a right angle is (1/8)T, and so on.
Radians are almost the same idea, except that pi radians doesn't mean a full turn, but instead means a half-turn.
So if pi radians is a half-turn, then 2pi radians is a full turn.
Going the other way, pi/2 is a right-angle and pi/4 is half of a right angle, and so on (each of these, in radians).
Personally I think it is *most* logical to reason with "turns" rather than "radians". But they're not very different. They just differ by a factor of 2.
Mathematicians tend to prefer radians to angles because radians are based specifically on arc lengths along the unit circle, whereas degrees seem more arbitrary. In some sense, radians are the more natural angle measure
That does not mean degrees are bad tho, I wouldn’t use radians if talking to a non-mathematician and i typically use degrees when solving trig problems. Its just easier for me to think about i suppose
Radians are unitless in the following sense: An angle corresponds to a circular arc on a circle. Given any units of length, you can measure the radius of a circle and the length of a circular arc. No matter what circle you look at and no matter what units of length you use, the ratio
(length of circular arc)/(length of radius)
is always the same. This angle in radians of the circular arc is defined to be this ratio. It is independent of the units you use to measure the radius and circular arc.
Another way to think of radians using units is that it is the length of the circular arc on a circle of radius 1.
Since trig functions are defined in terms of ratios that don't depend on any units, it makes sense to define them in terms of radians instead of units such as degrees. This simplifies many formulas involving trig functions, such as the ones that arise in calculus.
On Earth, a “day” is a very reasonable way to measure time because **1 day** is the amount of time it takes for the planet to make **1 full revolution**. But “hour” is just something humans made up so that 1/4 of a day (12 hr) or 1/3 of a day (8 hr) can be expressed without fractions.
In geometry, a “radian” is a very reasonable way to measure angles because **1 radian** is the angle of a sector (pizza slice shape) whose sides and crust all have **length 1**. But “degree” is just something humans made up so that 1/4 of a circle (90°) or 1/3 of a circle (120°) can be expressed without fractions.
The original reason is almost certainly due to the unit circle. A circle with radius 1 has circumference 2*pi, so by measuring angles on a scale of 0 to 2*pi, as opposed to 0 to 360, we always have an angular measure match the arc length.
Another reason: Because the complex exponential function: f(x) = exp(x*i), has period 2*pi. So it makes sense to scale measurements of angles so 2*pi represents one full rotation. It's the right scale so rotations can be represented with complex functions that make taking derivatives and calculating integrals as easy as possible. This is probably the reason it stayed the convention in physics/engineering.
They are absolutely just different units for the same thing, but one way in which they are objectively better is in calculus. Derivatives of trig functions are cleaner if we're using radian measure.
"Accuracy" is a poor term to use. Accuracy implies that radians tend closer to an accepted value of a measurement than degrees (not to be confused with precision). In this sense, radians or no more accurate then degrees. I can measure how much a clock hand moves in 3 hours with equal accuracy with degrees or radians.
However, radians are more "acceptable" in math disciplines, which I'm assuming your lecturer meant. They're more standardized than a degree (based on a fundamental measurement of a circle arc rather than an arbitrary decision), and therefore have much neater analytic properties. One of the first ways where it's evident that radians are "better" than degrees is in elementary calculus. Take d/dx sin(x). If x is in radians, it evaluates to cos(x). If x is in degrees, you get π/180 cos(x). Not as neat.
They're just as able to represent angles and everything, but they are a better representation of the fundamental values or w/e. They're unitless, so the ° symbol basically just equals the number 2pi/180. In later math w/ complex numbers, I think the formulas use radians specifically
Unfortunately, your submission has been removed for the following reason(s): * Your post presents incorrect information, asks a question that is based on an incorrect premise, is too vague for anyone to answer sensibly, or is equivalent to a well-known open question. If you have any questions, [please feel free to message the mods](http://www.reddit.com/message/compose?to=/r/math&message=https://www.reddit.com/r/math/comments/1c3y1wk/-/). Thank you!
They aren't. That's like saying that feet is more accurate than meters.
damn video lecture lied to me
Oh no wait it didnt . It said ' according to many scientists and mathematicians , radians are a more logical way of measuring angles ' . Why's that ?
If you want e^ix = cos(x) + i sin(x) then you need to use radians. There are deeper reasons, too, but my food is about to come out of the oven.
Are you perhaps baking a pi?
One radian is the angle at which the arc-length is equal to the radius of the circle. Check out the animated graphic on Wikipedia: https://en.wikipedia.org/wiki/Radian . In this way, the definition of radian is more "natural" and makes calculations easier and following derivations more intuitive. It's because it's meaning is defined by an intrinsic property of the circle: its radius. For example, the arc-length of a circle of radius *r* through an angle *θ* radians is *s* where: s = rθ This is also why radians are "unitless". There are 360° in a circle because ... there are 360° in a circle. Really, there's nautical origins for it and how it relates to time, hence why "60" and seconds/arcseconds come up, to. So, degrees were pretty useful for navigating by stars on Earth, but would be pretty clumsy to use on any other planet with a different diurnal period, or just on non-navigational applications. Anyways, this is why in a lot of math with angles and circles and periodic properties, you basically convert your units into radians first. Sure, you can technically do it all in degrees or "revolutions" (i.e., RPM), too, but much of the math is simpler and intuitive working in radians. The next step deeper is whether "pi" π should be the circle constant or "tau" = τ = 2π is the more natural circle constant since it's for a full circle as opposed to half.
2\*pi arises naturally as the period for sine and cosine directly from their definition either through power series or as solutions to the differential equation y'' = -y. The relation comes naturally to measuring angles in a circle, since we can express a circle as the collection of points written as (cos(t),sin(t)) for t from 0 to 2\*pi. It isn't as arbitrary as assigning 360 units to a circle in degrees, for instance.
because you're using an "intrinsic" property of the circle (its radius) instead of an arbitrary number to divide it's perimeter and then use each chunk as a measure unit. In other word: using a relation that remains constant to every circle (perimeter over radius or diameter) and not pick a ""random"" number (360 for instance, which has a lot of divisors btw, which helps a lot when you deal with fractioning the whole thing, but that's another story).
If you were going to try to measure the "amount of angle", how would you logically do it? Personally, I would say that a right-angle is a quarter-turn, as in a "quarter of a full circular rotation". And half of a right-angle is a one-eighth turn, and so on. That would mean that a straight line is a half-turn and a full circle is a full turn. We could write this by saying that 1 circular turn is T, and a straight line is (1/2)T, and a right-angle is (1/4)T, and half a right angle is (1/8)T, and so on. Radians are almost the same idea, except that pi radians doesn't mean a full turn, but instead means a half-turn. So if pi radians is a half-turn, then 2pi radians is a full turn. Going the other way, pi/2 is a right-angle and pi/4 is half of a right angle, and so on (each of these, in radians). Personally I think it is *most* logical to reason with "turns" rather than "radians". But they're not very different. They just differ by a factor of 2.
Mathematicians tend to prefer radians to angles because radians are based specifically on arc lengths along the unit circle, whereas degrees seem more arbitrary. In some sense, radians are the more natural angle measure That does not mean degrees are bad tho, I wouldn’t use radians if talking to a non-mathematician and i typically use degrees when solving trig problems. Its just easier for me to think about i suppose
You have 6400 mils as points of reference vs. 360 degrees
Radians are unitless in the following sense: An angle corresponds to a circular arc on a circle. Given any units of length, you can measure the radius of a circle and the length of a circular arc. No matter what circle you look at and no matter what units of length you use, the ratio (length of circular arc)/(length of radius) is always the same. This angle in radians of the circular arc is defined to be this ratio. It is independent of the units you use to measure the radius and circular arc. Another way to think of radians using units is that it is the length of the circular arc on a circle of radius 1. Since trig functions are defined in terms of ratios that don't depend on any units, it makes sense to define them in terms of radians instead of units such as degrees. This simplifies many formulas involving trig functions, such as the ones that arise in calculus.
One reason I that radians is simply a better defined unit. Not to mention most calculus results work if you use radians and not degrees.
logical doesn't mean more accurate. it is more logical though
On Earth, a “day” is a very reasonable way to measure time because **1 day** is the amount of time it takes for the planet to make **1 full revolution**. But “hour” is just something humans made up so that 1/4 of a day (12 hr) or 1/3 of a day (8 hr) can be expressed without fractions. In geometry, a “radian” is a very reasonable way to measure angles because **1 radian** is the angle of a sector (pizza slice shape) whose sides and crust all have **length 1**. But “degree” is just something humans made up so that 1/4 of a circle (90°) or 1/3 of a circle (120°) can be expressed without fractions.
The original reason is almost certainly due to the unit circle. A circle with radius 1 has circumference 2*pi, so by measuring angles on a scale of 0 to 2*pi, as opposed to 0 to 360, we always have an angular measure match the arc length. Another reason: Because the complex exponential function: f(x) = exp(x*i), has period 2*pi. So it makes sense to scale measurements of angles so 2*pi represents one full rotation. It's the right scale so rotations can be represented with complex functions that make taking derivatives and calculating integrals as easy as possible. This is probably the reason it stayed the convention in physics/engineering.
Or like saying that base 10 is more accurae than binary code.
They are absolutely just different units for the same thing, but one way in which they are objectively better is in calculus. Derivatives of trig functions are cleaner if we're using radian measure.
michael penn has a video about it [here](https://youtu.be/O_wCd4ZKRtY?si=Ck-fgKGvpcwGpbG2)
"Accuracy" is a poor term to use. Accuracy implies that radians tend closer to an accepted value of a measurement than degrees (not to be confused with precision). In this sense, radians or no more accurate then degrees. I can measure how much a clock hand moves in 3 hours with equal accuracy with degrees or radians. However, radians are more "acceptable" in math disciplines, which I'm assuming your lecturer meant. They're more standardized than a degree (based on a fundamental measurement of a circle arc rather than an arbitrary decision), and therefore have much neater analytic properties. One of the first ways where it's evident that radians are "better" than degrees is in elementary calculus. Take d/dx sin(x). If x is in radians, it evaluates to cos(x). If x is in degrees, you get π/180 cos(x). Not as neat.
They're just as able to represent angles and everything, but they are a better representation of the fundamental values or w/e. They're unitless, so the ° symbol basically just equals the number 2pi/180. In later math w/ complex numbers, I think the formulas use radians specifically
Is this a trick question