Radians

Radian

From Wikipedia, the free encyclopedia

An angle of 1 radian results in an arc with an equal length to the radius of thecircle.

The radian is a unit of plane angle, equal to 180/π (or 360/(2π)) degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematicsbeyond the elementary level.

The radian 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 as ” 1.2 rad ” or ” 1.2c ” (the second symbol is often mistaken for a degree: ” 1.2° “). However, the radian is mathematically considered a “pure number” that needs no unit symbol, and in mathematical writing the symbol “rad” is almost always omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol ° is used.

The radian was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit. The SI unit of solid angle measurement is the steradian.

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[edit]Definition

One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s /r, where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = .

It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr /r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.

[edit]History

The concept of radian measure, as opposed to the degree of an angle, should probably be credited to Roger Cotes in 1714.[1] He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure.

The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson (brother of Lord Kelvin) at Queen’s CollegeBelfast. He used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, vacillated between radradial and radian. In 1874, Muir adopted radian after a consultation with James Thomson.[2][3][4]

[edit]Conversions

[edit]Conversion between radians and degrees

A chart to convert between degrees and radians

As stated, one radian is equal to 180/π degrees. Thus, to convert from radians to degrees, multiply by 180/π. For example,

 {deg} = {rad} \cdot \frac {180^\circ} {\pi}

For Example:

1 \mbox{ rad} = 1 \cdot \frac {180^\circ} {\pi} \approx 57.2958^\circ
2.5 \mbox{ rad} = 2.5 \cdot \frac {180^\circ} {\pi} \approx 143.2394^\circ
\frac {\pi} {3} \mbox{ rad} = \frac {\pi} {3} \cdot \frac {180^\circ} {\pi} = 60^\circ

Conversely, to convert from degrees to radians, multiply by π/180.

 {rad} = {deg} \cdot \frac {\pi} {180^\circ}

For Example:

1^\circ = 1 \cdot \frac {\pi} {180^\circ} \approx 0.0175 \mbox{ rad}

23^\circ = 23 \cdot \frac {\pi} {180^\circ} \approx 0.4014 \mbox{ rad}

Radians can be converted to revolutions by dividing the number of radians by 2π.

[edit]Conversion between radians and grads

2π radians are equal to one complete revolution, which is 400g. So, to convert from radians to grads multiply by 200/π, and to convert from grads to radians multiply by π/200. For example,

1.2 \mbox{ rad} = 1.2 \cdot \frac {200^{\rm g}} {\pi} \approx 76.3944^{\rm g}
50^{\rm g} = 50 \cdot \frac {\pi} {200^{\rm g}} \approx 0.7854 \mbox{ rad}

The table shows the conversion of some common angles.

Units Values
Revolutions   0 1/12

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Special triangles


Fibonacci triangles

Starting with 5, every other Fibonacci number {0,1,1,2,3,5,8,13,21,34,55,89,…} is the length of the hypotenuse of a right triangle with integral sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.

The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely and approaches a limiting triangle with edge ratios:

\sqrt{5}:2:1.

This right triangle is sometimes referred to as a dom, a name suggested by Andrew Clarke to stress that this is the triangle obtained from dissecting a domino along a diagonal. The dom forms the basis of the aperiodic pinwheel tiling proposed by John Conway and Charles Radin.

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