Lesson RADIANS

Source code of 'RADIANS'

This Lesson (RADIANS) was created by by Theo(13342)

This lesson provides a brief overview of Radians.

REFERENCES

<a href = "http://en.wikipedia.org/wiki/Radian" target = "_blank">http://en.wikipedia.org/wiki/Radian</a>
<a href = "http://math.rice.edu/~pcmi/sphere/drg_txt.html" target = "_blank">http://math.rice.edu/~pcmi/sphere/drg_txt.html</a>
<a href = "http://www.themathpage.com/atrig/arc-length.htm" target = "_blank">http://www.themathpage.com/atrig/arc-length.htm</a>
<a href = "http://www.clarku.edu/~djoyce/trig/angle.html" target = "_blank">http://www.clarku.edu/~djoyce/trig/angle.html</a>
<a href = "http://www.absoluteastronomy.com/topics/Radian" target = "_blank">http://www.absoluteastronomy.com/topics/Radian</a>

TERMINOLOGY

In this lesson:

pi stands for the greek letter {{{pi}}}. 
r stands for radius of a circle.
C stands for circumference of a circle.
S stands for length of an arc.
A[r] stands for angle expressed in radians.
A[d] stands for angle expressed in degrees.
d by itself (not connected with A[d]) stands for diameter of a circle.
* indicates multiplication 
/ indicates division
() means do the mathematical operations within the parentheses first.
[] means sub as in A[d] means A sub degrees stands for angle expressed in degrees.

BASIC DEFINITION OF A RADIAN

A Central Angle of a Circle expressed in radians is the ratio of the Arc Length subtended by that angle to the radius of the circle.

The formula would be:

A[r] = S/r

where:

A[r] is the central angle of the circle expressed in radians.
S is the length of the arc on the circumference of the circle created by that angle.
r is the radius of the circle.

DEVELOPMENT OF THE CONCEPT OF RADIAN

Formula for the circumference of a circle is:

C = 2*pi*r

Where:

C = the circumference of the circle.
r = the radius of the circle.
pi = the constant of 3.141592654 which expresses the ratio of the Circumference of the Circle to the Diameter of the Circle.

Since the diameter of a circle is equal to 2 * the radius of a circle, the formula for the circumference of a circle can be either (2*pi*r) or (pi*d).

(2*pi) tells us the number of times we have to multiply the radius in order to find the length of the arc of a complete circle.   This represents the length of an arc created by an angle of 360 degrees.

If we wanted to find the length of the arc of a half a circle, we would take half of this.

Our formula would be C/2 = (1/2) * 2 * pi * r which would become pi * r.

(pi) tells us the number of times we have to multiply the radius in order to find the length of the arc of a half of a circle.  This represents the length of an arc created by an angle of 180 degrees.

If we wanted to find the length of the arc of a quarter of a circle, we would take one fourth of the circumference of the circle.

Our formula would be C/4 = (1/4) * 2 * pi * r which would become (pi/2) * r.

(pi/2) tells us the number of times we have to multiply the radius in order to find the length of the arc of a quarter of a circle.  This represents the length of an arc created by an angle of 90 degrees.

RELATIONSHIP BETWEEN THE ANGLE AND THE RADIUS

An arc length of:

360 degrees = (2*pi)*r
180 degrees = (pi)*r
90 degrees = (pi/2) * r

This was derived from the formula that says:

S = (A[d]/360) * 2 * pi * r where:

S = Arc Length
A[d] = Angle in degrees

For the full circle we get S = (360/360) * 2 * pi * r =  2 * pi * r = (2*pi) * r.
For the half circle we get S = (180/360) * 2 * pi * r = (1/2) * 2 * pi * r = pi * r = (pi) * r.
For the quarter circle circle we get S = (90/360) * 2 * pi * r = (1/4) * 2 * pi * r = (1/2) * pi * r = (pi/2) * r.

This led to the idea that if 360 degrees was expressed as (2*pi), and 180 degrees was expressed as (pi), and 90 degrees was expressed as (pi/2), then formulas involving lengths of arcs could be expressed directly from the angle rather than having to do a conversion each time.

This led to the creation of the radian as a unit of measure for an angle.

This was a very important mathematical development.

The use of radians facilitated the development of mathematical formulas in Calculus dealing with trigonometric functions, and other areas.

The listed references will provide you with a much more detailed and thorough analysis of the history of the development of radians as a unit of measure for angles.  If you haven't already done so, please take the time to look at them.

The basic idea was that working with radians rather than degrees simplified formulas making them much easier to work with.

Radians were considered a more natural expression of the measure of the angle rather than degrees.

With radians, the angle is expressed as a ratio of the length of the arc divided by the radius.

This has become the international standard for the measure of an angle.

You are usually taught the basics of trigonometry in degrees.

As you advance in mathematics and the sciences, radians become more prominent and universally used.

CONVERSION FROM DEGREES TO RADIANS

The formula to convert from degrees to radians is this:

A[r] = (A[d] * (pi/180)

If A[d] = 360, then A[r] = 360 * (pi/180) = 2*pi
If A[d] = 180, then A[r] = 180 * (pi/180) = pi
If A[d] = 90, then A[r] = 90 * (pi/180) = pi/2

The formula for the length of an arc becomes:

S = A[r] * r

which is much simpler than:

S = (A[d]/360) * 2 * pi * r

You get the same answer in a more direct way.

Assume your angle is 30 degrees.

Using degrees, the length of the arc would be equal to:

S = (A[d]/360) * 2 * pi * r = (30/360) * 2 * pi * r = (1/12) * 2 * pi * r = (2/12) * pi * r = (1/6) * pi * r = (pi/6) * r

Using radians, the length of the arc would be equal to:

S = A[r] * r = (pi/6) * r

This is because:

If A[d] = 30 degrees, then A[r] = 30 * (pi/180) = (pi/6)

MEASURE OF ONE DEGREE IN RADIANS

The formula to convert from degrees to radians is:

number of degrees = number of radians * (180 / pi)

If number of degrees equals 1, this formula becomes:

1 = number of radians * (180 / pi)

Solve for number of radians to get:

number of radians = (pi / 180)

Since pi = 3.141592654, then number of radians = .017453293 radians

This means that 1 degree = .017453293 radians

MEASURE OF ONE RADIAN IN DEGREES

The formula to convert from radians to degrees is:

number of radians = number of degrees * (pi / 180)

If the number of radians is equal to 1, this formula becomes:

1 = number of degrees * (pi / 180)

Solve for number of degrees to get:

number of degrees = (180 / pi)

Since pi = 3.141592654, then number of degrees = 57.29577951 degrees.

This means that 1 radian = 57.29577951 degrees.

You have:

1 Degree = .017453293 radians
1 Radian = 57.29577951 degrees.

SUMMARY

A Radian is a measure of the angle that subtends an arc with a length of 1 radius.

The circumference of a circle is the length of an arc for an angle of 360 degrees.  The formula is C = 2 * pi * r.   A 360 degree angle is expressed as 2 * pi radians.   The formula for the circumference of a circle becomes A[r] * r.   A[r] is the angle of 360 degrees expressed in radians.

An angle can be expressed in degrees or radians.

If in degrees, then the degree symbol is used.  The degree symbol looks like a super-scripted "o".

If in radians, then no symbol is used.  The angle without the degree symbol is assumed to be in radians.

If a radian symbol is used, then that symbol looks like a super-scripted "c", but is more commonly the term "rad" or "rads", as in 360 degrees = 2 * pi rads.

To convert from degrees to radians, you take your degrees and multiply them by (pi/180).

To convert from radians to degrees, you take your radians and multiply them by (180/pi)

The formula for the arc length of an angle in degrees is:

S = (A[d]/360) * 2 * pi * r

The formula for the arc length of an angle in radians is:

S = A[r] * r

A TABLE OF SOME WELL KNOWN ANGLES

<pre>
ANGLE IN DEGREES          ANGLE IN RADIANS

30                             pi/6
45                             pi/4
60                             pi/3
90                             pi/2
180                            pi
360                            2*pi
</pre>