what is a radian?

Question

From what I understand, a radian is a form of angle measure where it has 1 radian if the length of the radius is equal to the length of arc. But here's what confuses me, there are approx. 6.28 radians in a circle, but what if there is an arc created by a 170 degree angle. then there can be no more than 2 radians? Dammit I dont know my trig book really sucks. Someone please EXPLAIN to me radians clearly. Also what does subtend mean? I asked that earlier but I got 6 different anwers? Does it mean limited or connected to or what?

yea the first guy i know what a radian is but i jsut dont understand it you know? 2pi = 360 but i want more than that.

Answer 1

The radian is a unit of plane angle. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written "1.2 rad" or "1.2c".
Definition

The angle subtended at the center of a circle by an arc of circumference equal in length to the radius of the circle is one radian.

1 rad = m·m–1 = 1

In terms of a circle it can be seen as the ratio of the length of the arc subtended by two radii to the radius of the circle.
History

The term radian first appeared in print on June 5, 1873, in examination questions set by James Thomson at Queen's College, Belfast. James Thomson was a brother of Lord Kelvin. He used the term as early as 1871, while in 1869 Thomas Muir, then of St. Andrew's University, hesitated between rad, radial and radian. In 1874, Muir adopted radian after a consultation with James Thomson. (Sources: Florian Cajori, 1929, History of Mathematical Notations, Vol. 2, pp. 147-148; Nature, 1910, Vol. 83, pp. 156, 217, and 459-460; [1]).

The concept of a radian measure, as opposed to the degree of an angle, should probably be credited to Roger Cotes in 1714 [2]. He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure.

Answer 2

No, an arc created by a 170° angle would be approximately 2.96706 radians. Basically, if you have a circle with a center at the vertex of the angle, the measure of the angle in radians is the length of the arc inside the angle (in technical terms, the arc subtended by the angle) divided by the radius of the same circle. To convert between radians and degrees, note that each degree is 1/360 of the circle, and the total circumference of the circle is 2πr, thus the measure in radians is θ/360*2π, which being simplified is πθ/180 (assuming θ is in degrees). The usefulness of radians is something that won't become obvious until you reach calculus, but basically has to do with the fact that the derivative of sin x is cos x if and only if x is in radians. Further, the taylor series expansion for sin x = x - x^3/3! + x^5/5! - x^7/7! + x^9/9!... is valid if and only if x is in radians, so having the angle in radians allows you to calculate angles analytically instead of geometrically, which is usually MUCH faster.

Subtend literally means to extend under, but in the context of circles, it's always the part of the circle that is between the points where the angle/chord/whatever intersects the circle.

Answer 3

Subtend means the angle formed by drawing two radii, one from each end of an arc. So a quarter of a circle subtends an angle of 90 degrees. The best way to think about radians is that it's another method to measure angles, like meter and yard are different ways to measure length. The conversion fact is pi radians (3.14 radians) = 180 degrees. So your 170 degree angle would convert to slightly less than 3.14 radians.

Answer 4

The radian is a unit of plane angle. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written "1.2 rad" or "1.2c ".
The radian was formerly an SI supplementary unit, but this category was abolished from the SI system in 1995. For measuring solid angles, see steradian.
Nowadays, radian is the de facto unit of plane angles for mathematicians, and the symbol "rad" is usually omitted in mathematicial writings. When using degrees, the ° symbol is used to distinguish it from radians.
One radian is the angle subtended at the center of a circle by an arc of circumference that is equal in length to the radius of the circle.


In terms of a circle it can be seen as the ratio of the length of the arc subtended by two radii to the radius of the circle.
For more please visit:
http://en.wikipedia.org/wiki/Radian

Answer 5

1 radian = 1 distance unit of arc length for a circle with a unit radius

where arc length is measured for the angle subtended at the centre by the same chord.

Radian measure is true circular measure. What this means is that there are exactly 2*pi radians in a complete angle of revolution. This is untrue for degrees, i.e. a circle can have any number of degrees you wish. e.g. it is quite valid to have a circle with 240 degrees.

All angles in a circle are subtended on chords. This means the arms of each angle terminate in the opposite ends of a chord. The arc formed by this chord is the arc length used in calculating radians.

It is untrue that we would not be able to find the derivative of sin x if x were not in radians as Pascal claims. It is certainly easier however.

Answer 6

Subtend: to extend under or be opposite : a cord subtends and arc. (The Random House College Dictionary)

With regard to radian, think of two different circles of different sizes, therefore the radius of each circle is of a different size.

If you were to cut a piece of string that was the length of the radius of the smaller circle, and then lay that string along the perimeter of the smaller circle, and then mark the end pts of that string on the circle, and then draw a cord from those end pts, then draw line segments from the center of the circle to those end points, the central angle would be 1 radian.

Note if you did the same thing to the other larger circle but used the same string segment, you would not have a radian.

Note: Radian = length along perimeter of circle/radius.

You are not restricted to the direction you go (counter clock wise is positive, clockwise is negative) or how far you go.

If you have a race track that is circular, and has a 1 mile radius and you drive counterclockwise 250 time how many radians did you go? how many miles did you go?

Answer 7

2

Answer 8

A radian is a ratio of lengths. Walk along the edge of any circle, divide the distance that you've walked by the radius of the circle and that is the measure of the angle in radians.
If you walk all the way around the circumference you've walked
c=2*pi*r divide by the radius, r, to get that angle of
a complete circle, is 2 pi = 6.28 radians.

You now know that a complete circle is 360 degrees or 2pi radians. This gives you a conversion factor for any other angle:

170 deg * pi/180 = 2.97 radians.

Once you get used to radians it becomes natural to refer to parts of circles in terms of pi:

A complete circle is 2 pi,
180 degrees, of a half circle is pi,
90 degrees, or quarter circle is pi/2.
45 degrees is pi/4
30 deg is pi/6
etc.
Notice how these common angles have such a nice form.

The radian is the natural way to measure angles.

Answer 9

You give the correct description of a radian:
The angle measured in radians is the lenght of the arc divided by the radius.

The easiest way understand radians is to assume the radius is one.
The whole circle now has a perimeter of 2*pi = 6.28
Therefore 360 deg corresponds to about 6.28 radians

=> 180 deg corresponds to pi or about 3.14 radians
A straight corner of 90 deg is pi/2 or about 1.57 radians


Given the angle in degrees you can calculate the angle in radians by multiplying with pi and dividing by 180:
e.g. 170 deg = 170 * pi/180 = 2,97 radians

Given the angle in radians you can calculate the angle in degrees by multiplying with 180 and dividing by pi:
e.g. 1 rad = 180/pi = 57,3 deg

Answer 10

Think of cutting out a piece of pie. Start with the knife in the center and cut towards the edge.That is one radian.Now go back to the center and cut the second line.That is the second radian.Those two radians have subtended or cut out your piece of pie.

Subtend means to set, mark, or draw the boundaries of something.

That will be the day math is for everyone when the teachers stop with the enigmatic terms and explain it like a normal human being. What is with all the secret terms and formulas? In the Pythagorean era math was kept a secret and was devoted to secrecy.I dont think we have ever truly recovered from that mode of thinking concerning mathmatics.