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 Chapter 1\

 The tao

 that can

 be described

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 The name that can be spoken

 is not the eternal Name

 The nameless is the boundary

 of Heaven and Earth

 The named

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Freed from desire

 you can see

 the hidden mystery

 By having desire

 you can only see

 what is visibly real

...Yet mystery and reality

 emerge

 from the same source

 This source is called

 darkness Darkness born

 from darkness

 The beginning of all understanding

Chapter 81

 True words

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; beautiful sounding words

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. Wise men don't need to debate

; men who need to debate

 are not wise

. Wise men are not scholars

, and scholars are not wise

. The Master

 desires

 no possessions

. Since the things

 she does

 are for

 the people

, she has

 more than she needs

. The more she gives to others

, the more she has for herself

. The Tao of Heaven nourishes

 by not forcing

. The Tao

 of the Wise person

 acts

 by not competing

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Metronomic rock sundial speaks clock

...

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Cobalt Blue picture of you

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2

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4

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180

360

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do the eyeballs bounce

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Radian

From Wikipedia, the free encyclopedia

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For a measure of ionizing radiation, see Rad (unit). For other uses, see Radian (disambiguation).

Radian
Unit system	SI derived unit
Unit of	Angle
Symbol	rad, c or r
Conversions
1 rad in ...	... is equal to ...
milliradians	1000 mrad
turns	1/2π turn
degrees	180°/π ≈ 57.296°
gradians	200g/π ≈ 63.662g

An arc of a circle with the same length as the radius of that circle subtends an angle of 1 radian. The circumference subtends an angle of 2π radians.

The radian, denoted by the symbol rad, is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that category was abolished in 1995) and the radian is now an SI derived unit.^[1] The radian is defined in the SI as being a dimensionless unit with 1 rad = 1.^[2] Its symbol is accordingly often omitted, especially in mathematical writing.

Contents

• 1Definition
• 2Unit symbol
• 3Conversions
  • 3.1Conversion between radians and degrees
    • 3.1.1Radian to degree conversion derivation
  • 3.2Conversion between radians and gradians
• 4Advantages of measuring in radians
• 5History
  • 5.1History as an SI unit
• 6Dimensional analysis
• 7Use in physics
• 8SI multiples
• 9See also
• 10Notes
• 11References
• 12External links

Definition

One radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle.^[3] More generally, the magnitude in radians of 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. A right angle is exactly π/2 radians.^[4]

A complete revolution is 2π radians (shown here with a circle of radius one and thus circumference 2π).

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 ≈ 57.295779513082320876 degrees.^[5]

The relation 2π rad = 360° can be derived using the formula for arc length, ${\displaystyle \ell _{\text{arc}}=2\pi r\left({\tfrac {\theta }{360^{\circ }}}\right)}$, and by using a circle of radius 1. Since radian is the measure of an angle that subtends an arc of a length equal to the radius of the circle, ${\displaystyle 1=2\pi \left({\tfrac {1{\text{ rad}}}{360^{\circ }}}\right)}$. This can be further simplified to ${\displaystyle 1={\tfrac {2\pi {\text{ rad}}}{360^{\circ }}}}$. Multiplying both sides by 360° gives 360° = 2π rad.

Unit symbol

The International Bureau of Weights and Measures^[4] and International Organization for Standardization^[6] specify rad as the symbol for the radian. Alternative symbols used 100 years ago are ^c (the superscript letter c, for "circular measure"), the letter r, or a superscript ^R,^[7] but these variants are infrequently used, as they may be mistaken for a degree symbol (°) or a radius (r). Hence a value of 1.2 radians would most commonly be written as 1.2 rad; other notations include 1.2 r, 1.2^rad, 1.2^c, or 1.2^R.

In mathematical writing, the symbol "rad" is often omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant, the degree sign ° is used.

Conversions

A chart to convert between degrees and radians

Conversion of common angles

Turns	Radians	Degrees	Gradians, or gons
0 turn	0 rad	0°	0g
1/24 turn	π/12 rad	15°	16+2/3g
1/16 turn	π/8 rad	22.5°	25g
1/12 turn	π/6 rad	30°	33+1/3g
1/10 turn	π/5 rad	36°	40g
1/8 turn	π/4 rad	45°	50g
1/2π turn	1 rad	c. 57.3°	c. 63.7g
1/6 turn	π/3 rad	60°	66+2/3g
1/5 turn	2π/5 rad	72°	80g
1/4 turn	π/2 rad	90°	100g
1/3 turn	2π/3 rad	120°	133+1/3g
2/5 turn	4π/5 rad	144°	160g
1/2 turn	π rad	180°	200g
3/4 turn	3π/2 rad	270°	300g
1 turn	2π rad	360°	400g

Conversion between radians and degrees

As stated, one radian is equal to ${\displaystyle {180^{\circ }}/{\pi }}$. Thus, to convert from radians to degrees, multiply by ${\displaystyle {180^{\circ }}/{\pi }}$.

${\displaystyle {\text{angle in degrees}}={\text{angle in radians}}\cdot {\frac {180^{\circ }}{\pi }}}$

For example:

${\displaystyle 1{\text{ rad}}=1\cdot {\frac {180^{\circ }}{\pi }}\approx 57.2958^{\circ }}$

${\displaystyle 2.5{\text{ rad}}=2.5\cdot {\frac {180^{\circ }}{\pi }}\approx 143.2394^{\circ }}$

${\displaystyle {\frac {\pi }{3}}{\text{ rad}}={\frac {\pi }{3}}\cdot {\frac {180^{\circ }}{\pi }}=60^{\circ }}$

Conversely, to convert from degrees to radians, multiply by ${\displaystyle {\pi }/{180^{\circ }}}$.

${\displaystyle {\text{angle in radians}}={\text{angle in degrees}}\cdot {\frac {\pi }{180^{\circ }}}}$

For example:

${\displaystyle 1^{\circ }=1^{\circ }\cdot {\frac {\pi }{180^{\circ }}}\approx 0.0175{\text{ rad}}}$

${\displaystyle 23^{\circ }=23^{\circ }\cdot {\frac {\pi }{180^{\circ }}}\approx 0.4014{\text{ rad}}}$

Radians can be converted to turns (complete revolutions) by dividing the number of radians by 2π.

Radian to degree conversion derivation

The length of circumference of a circle is given by ${\displaystyle 2\pi r}$, where ${\displaystyle r}$ is the radius of the circle.

So the following equivalent relation is true:

${\displaystyle 360^{\circ }r\iff 2\pi r}$ [Since a ${\displaystyle 360^{\circ }}$ sweep is needed to draw a full circle]

By the definition of radian, a full circle represents:

${\displaystyle {\frac {2\pi r}{r}}{\text{ rad}}}$

${\displaystyle =2\pi {\text{ rad}}}$

Combining both the above relations:

${\displaystyle 2\pi {\text{ rad}}=360^{\circ }}$

${\displaystyle \Rrightarrow 1{\text{ rad}}={\frac {360^{\circ }}{2\pi }}}$

${\displaystyle \Rrightarrow 1{\text{ rad}}={\frac {180^{\circ }}{\pi }}}$

Conversion between radians and gradians

${\displaystyle 2\pi }$ radians equals one turn, which is by definition 400 gradians (400 gons or 400^g). So, to convert from radians to gradians multiply by ${\displaystyle 200^{\text{g}}/\pi }$, and to convert from gradians to radians multiply by ${\displaystyle \pi /200^{\text{g}}}$. For example,

${\displaystyle 1.2{\text{ rad}}=1.2\cdot {\frac {200^{\text{g}}}{\pi }}\approx 76.3944^{\text{g}}}$

${\displaystyle 50^{\text{g}}=50^{\text{g}}\cdot {\frac {\pi }{200^{\text{g}}}}\approx 0.7854{\text{ rad}}}$

Advantages of measuring in radians

Some common angles, measured in radians. All the large polygons in this diagram are regular polygons.

In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians. This is because radians have a mathematical "naturalness" that leads to a more elegant formulation of a number of important results.

Most notably, results in analysis involving trigonometric functions can be elegantly stated, when the functions' arguments are expressed in radians. For example, the use of radians leads to the simple limit formula

${\displaystyle \lim _{h\rightarrow 0}{\frac {\sin h}{h}}=1,}$

which is the basis of many other identities in mathematics, including

${\displaystyle {\frac {d}{dx}}\sin x=\cos x}$^[5]

${\displaystyle {\frac {d^{2}}{dx^{2}}}\sin x=-\sin x.}$

Because of these and other properties, the trigonometric functions appear in solutions to mathematical problems that are not obviously related to the functions' geometrical meanings (for example, the solutions to the differential equation ${\displaystyle {\tfrac {d^{2}y}{dx^{2}}}=-y}$, the evaluation of the integral ${\displaystyle \textstyle \int {\frac {dx}{1+x^{2}}},}$ and so on). In all such cases, it is found that the arguments to the functions are most naturally written in the form that corresponds, in geometrical contexts, to the radian measurement of angles.

The trigonometric functions also have simple and elegant series expansions when radians are used. For example, when x is in radians, the Taylor series for sin x becomes:

${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots .}$

If x were expressed in degrees, then the series would contain messy factors involving powers of π/180: if x is the number of degrees, the number of radians is y = πx / 180, so

${\displaystyle \sin x_{\mathrm {deg} }=\sin y_{\mathrm {rad} }={\frac {\pi }{180}}x-\left({\frac {\pi }{180}}\right)^{3}\ {\frac {x^{3}}{3!}}+\left({\frac {\pi }{180}}\right)^{5}\ {\frac {x^{5}}{5!}}-\left({\frac {\pi }{180}}\right)^{7}\ {\frac {x^{7}}{7!}}+\cdots .}$

In a similar spirit, mathematically important relationships between the sine and cosine functions and the exponential function (see, for example, Euler's formula) can be elegantly stated, when the functions' arguments are in radians (and messy otherwise).

We will skip the history section for today because some hash is better

scraped down the hash hole

hot before habanero

heads hasta manna

head space ward