Circular System

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The angles considered in Euclidean Geometry are all less than two right angles, but for the purpose of Trigonometry it is necessary to extend the conception of angular magnitude so as to include angles of all magnitudes, positive or negative.

Suppose that the straight line $OP$ in the figure is capable of revolving about the point $O$, and suppose that in this way it has passed successively from the position $OA$ to the positions occupied by $OB$, $OC$, $OD$, $\ldots$, then the angle between $OA$ and any position such as $OC$ is measured by the amount of revolution which the line $OP$ has undergone in passing from its initial position $OA$ into its final position $OC$. We denote this angle by $\angle AOC$.

The point $O$ is called the origin, $OA$ the initial line and $OC$ the terminal line; the revolving line $OP$ is known as the generating line or the radius vector. The lines $OA$ and $OC$ are called the boundary lines of the angle $\angle AOC$.

For a complete revolution $OP$ makes four right angles; and moreover, the line $OP$ may make any number of complete revolutions through the original position ...
... $OA$ before taking up its final position $OC$. It will thus be seen that in Trigonometry angles are not restricted as in Euclid, but may be of any magnitude.

Moreover, $OP$ may revolve about the point $O$, either in clockwise direction or counter-clockwise direction. We adopt the convention that :

When the revolution of the radius vector $OP$ is counter-clockwise, the angle measured is positive.
When the revolution of the radius vector $OP$ is clockwise, the angle measured is negative.

In order to measure angles, we must decide upon a unit angle, which may be any arbitrarily chosen angle of fixed magnitude; then all other angles will be measured numerically by the ratios they bear to this unit angle. The natural unit to take would be right angle, but as the angles of ordinary size would then be denoted by fractions less than unity, it is more convenient to take a smaller angle as the unit.

The three systems of measurement that have been established are :

Sexagesimal System of Measurement
Centesimal System of Measurement
Circular System of Measurement

Sexagesimal System of Measurement (The English System)

In the sexagesimal system of measurement, a right angle is divided into $90$ equal parts called Degrees. Each degree is divided into $60$ equal parts called Minutes, and each minute into $60$ equal parts called Seconds.

The symbols $1^\circ$, $1^\prime$, and $1^{\prime\prime}$ are used to denote a degree, a minute, and a second respectively.

Thus,

$60$ Seconds ($60^{\prime\prime}$) = One Minute ($1^\prime$),

$60$ Minutes ($60^\prime$) = One Degree ($1^\circ$), and

$90$ Degrees ($90^\circ$) = One Right Angle.

An angle is measured by stating the number of degrees, minutes, and seconds which it contains. Thus, the angle which contains $53$ degrees, $37$ minutes, $2.53$ seconds is expressed symbolically in the form $53^\circ\:37^\prime\:2.53^{\prime\prime}$.

This system is well established and is always used in the practical applications of Trigonometry. It is not however very convenient on account of the multipliers $60$ and $90$.

Centesimal System of Measurement (The French System)

In the centesimal system of measurement, a right angle is divided into $100$ equal parts, called Grades. Each grade is divided into $100$ equal parts called Minutes, and each minute into $100$ equal parts called Seconds

The symbols $1^g$, $1^\prime$, and $1^{\prime\prime}$ are used to denote a grade, a minute, and a second respectively.

Thus,

$100$ Seconds ($100^{\prime\prime}$) = One Minute ($1^\prime$),

$100$ Minutes ($100^\prime$) = One Grade ($1^g$), and

$100$ Grades ($100^g$) = One Right Angle.

In this system the angle which contains $63$ grades, $70$ minutes, and $30.51$ seconds is expressed symbolically as $63^g\:70^\prime\:30.51^{\prime\prime}$.

The centesimal system of measurement was proposed at the time of the French Revolution as part of a general system of decimal measurement. This system would be much more convenient to use than the ordinary Sexagesimal System. But, it has never been adopted even in France, as it would have made necessary the alteration of Geographical, Nautical, Astronomical, and other tables prepared according to the sexagesimal method.
Remark
It will be noticed that different accents are used to denote sexagesimal and centesimal minutes and seconds; for though they have the same names, a centesimal minute and second are not the same as a sexagesimal minute and second. Thus a right angle contains $90 \times 60 = 5,400$ sexagesimal minutes, whereas it contains $100 \times 100 = 10,000$ centesimal minutes.

Circular System of Measurement

In the circular system of measurement, angles are not measured in terms of a submultiple of the right angle, as in the sexagesimal and centesimal systems of measurements, but a certain angle known as a $\text{radian}$ is taken as the standard unit, in terms of which all other angles are measured.

A radian is the angle subtended at the centre of any circle by an arc equal in length to the radius of the circle and is denoted by $1^c$.

In the figure, a cirlce with centre $O$ and radius $r$ is taken. If on the circumference, we measure an arc $\stackrel{\frown}{AB}$ equal to the radius and join $\overline{OA}$, $\overline{OB}$, the angle $\angle AOB$ is a radian.

In any system of measurement, it is essential that the unit should be always the same. In order to shew that a radian, constructed according to the above definition, is of constant magnitude, we use the following two propositions are from the Euclid's Elements :
Euclid's Elements, Book VI, Proposition 4

In equiangular triangles the sides about the equal angles are proportional, and those are corresponding sides whch subtend the equal angles.
Euclid's Elements, Book VI, Proposition 33
In equal cirlces, angles have the same ratio as the circumference on which they stand, whether they stand at the centers or at the circumference.

We use these two propositions in establishing the following theorem and then to prove the fact that a radian is of constant magnitude.

Theorem : The length of the circumference of a circle always bears a constant ratio to its diameter.

Take any two circles $S$ and $T$ whose radii are $r_1$ and $r_2$, and in each circle let a regular polygon of $n$ sides be described.

Let $\overline{A_1B_1}$ be a side of the first, $\overline{A_2B_2}$ a side of the second polygon, and let their lengths be deonted by $a_1$ and $a_2$. Join their extremities to $O_1$ and $O_2$ the centres of the circles. We thus obtain two isosceles triangles whose vertical angles are equal, each being $\frac{1}{n}$ of four right angles.

Hence the triangles are equiangular, and therefore we have by $\text{Euclid's Elements, Book VI, Proposition 4}$,

$\frac{A_1B_1}{O_1A_1} = \frac{A_2B_2}{O_2A_2}$;

that is, $\frac{a_1}{r_1} = \frac{a_2}{r_2}$;

$\therefore \quad \frac{na_1}{r_1} = \frac{na_2}{r_2}$;

that is, $\frac{p_1}{r_1} = \frac{p_2}{r_2}$,

where $p_1$ and $p_2$ are the perimeters of the polygons. This is true whatever be the number of sides in the polygons. Let then the number of sides be indefinitely increased (i.e., let $n$ become inconveniently great) so that finally the perimeters of the two polygons differ from the circumferences of the corresponding circles by as small a quantity as we please; so that ultimately

$\frac{c_1}{r_1} = \frac{c_2}{r_2}$,

where $c_1$ and $c_2$ are the circumferences of the two circles.

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It thus appears that the ratio of the circumference of a circle to its radius is the same whatever be the size of the circle; that is, in all circles

$\frac{Circumference}{Diameter}$

is a constant quantity.

This constant is incommensurable and is always denoted by the Greek letter $\pi$ ( read as "pi").

Unfortunately the value of $\pi$ is not a whole number, nor can it be expressed in the form of a vulgar fraction, and hence not in the form of a decimal fraction, terminating or recurring.

The number $\pi$ is an incommensurable magnitude, i.e., a magnitude whose value cannot be exactly expressed as the ratio of two whole numbers.

Its value, corerct to $8$ decimal places is, $3.14159265\ldots$.

The fraction $\frac{22}{7}$ gives the value of $\pi$ correctly for the first two decimal places; for $\frac{22}{7} $ $= 3.14285\ldots$.

The fraction $\frac{355}{113} $ $= 3.14159203\ldots$ is a more accurate value of $\pi$, being correct to $6$ decimal places.

Now, for any circle with radius $r$ and circumference $C$, we have

$\frac{Circumference}{Diameter} = \pi$

$\Rightarrow \quad \frac{C}{2r} = \pi$

$\Rightarrow \quad C = 2\pi r.$

Circumference of a circle with radius $r = 2\pi r$.

The Radian is Constant Angle

Draw any circle with centre $O$ and radius $r$. Let $A$ be point on the circumference. Then $OA = r$. Let the arc $\stackrel{\frown}{AB}$ be measured equal in length to $\overline{OA}$. Join $OB$; then $\angle AOB$ is a radian. Produce $AO$ to meet the circumference in $C$.

By $\text{Euclid's Elements, Book VI, Proposition 33}$, angles at the center of a circle are proportional to the arcs on which they stand; hence

$\frac{\angle AOB}{two \; right\; angles}$ $=$ $\frac{arc\; AB}{arc\; ABC}$

$\Rightarrow \quad \frac{1\; radian}{two\; right\; angles}$ $=$ $\frac{r}{\pi r}$

$\Rightarrow \quad 1\; radian =$ $\frac{1}{\pi}$ $\times$ two right angles.

$\Rightarrow \quad 1$ radian $=$ constant.

Since a radian is constant, it is taken as a standard unit, and the number of radians contained in any angle is spoken of as its radian measure or circular measure.
From the above discussion, it is clear that
$\pi^c = $ Two right angles
$\Rightarrow \;\; $ $\frac{\pi}{2}^c$ $ = $ One right angle.

Conversion to and from Circular System of Measurement

We know that

One Right Angle $= 90^\circ = 100^g =$ $\frac{\pi}{2}^c$.

$\Rightarrow$ Two Right Angles $= 180^\circ = 200^g = \pi^c$

This is the basic identity from which we can convert an angle in one measure to another measure.
Conversion from Circular Measure to Sexagesimal Measure and to Centesimal Measure

Reducing $(\frac{\pi}{64})^c$ into Sexagesimal Measure and into Centesimal Measure
To Sexagesimal Measure
Since,
$\pi^c = 180^\circ$
we have
$\Rightarrow 1^c =$ $ (\frac{180}{\pi})^\circ$,
$\Rightarrow x^c =$ $ (\frac{180x}{\pi})^\circ$.

So,

$(\frac{\pi}{64})^c$ $=$ $(\frac{180 \times (\frac{\pi}{64})}{\pi})^\circ$

$=$ $(\frac{\frac{180}{64} \times \pi}{\pi})^\circ$

$=$ $ (\frac{2.8125 \times \pi}{\pi})^\circ$

$=2.8125^\circ$

$=2^\circ + (0.8125 \times 60)^\prime$

$=2^\circ + 48.75^\prime$

$=2^\circ + 48^\prime + (0.75 \times 60)^{\prime\prime}$

$=2^\circ + 48^\prime + 45^{\prime\prime}$

$=2^\circ\;48^\prime\;45^{\prime\prime}.$
To Centesimal Measure
Since,
$\pi^c = 200^g$
we have
$\Rightarrow 1^c =$ $(\frac{200}{\pi})^g$,
$\Rightarrow x^c =$ $(\frac{200x}{\pi})^g$.

So,

$(\frac{\pi}{64})^c =$ $(\frac{200 \times (\frac{\pi}{64})}{\pi})^g$

$=$ $(\frac{\frac{200}{64} \times \pi}{\pi})^g$

$=$ $(\frac{3.125 \times \pi}{\pi})^g$

$=3.125^g$

$=3^g + (0.125 \times 100)^\prime$

$=3^g + 12.5^\prime$

$=3^g + 12^\prime + (0.5 \times 100)^{\prime\prime}$

$=3^g + 12^\prime + 50^{\prime\prime}$

$=3^g\;12^\prime\;50^{\prime\prime}.$
Conversion from Sexagesimal Measure to Circular Measure
Since
$180^\circ = \pi^c$
we have
$\Rightarrow 1^\circ =$ $(\frac{\pi}{180})^c$,
$\Rightarrow x^\circ =$ $(\frac{\pi x}{180})^c$.

For example,

$12^\circ\;13^\prime\;14.34^{\prime\prime} = 12.22065^\circ$

$=$ $(\frac{\pi \times 12.22065}{180})^c$

$=0.0678925\;\pi^c$
Conversion from Centesimal Measure to Circular Measure
Since,
$200^g = \pi^c$
we have
$\Rightarrow 1^g =$ $(\frac{\pi}{200})^c$,
$\Rightarrow x^g =$ $(\frac{\pi x}{200})^c$.

For example,

$13^g\;57^\prime\;85^{\prime\prime} = 13.5785^g$

$=$ $(\frac{\pi \times 13.5785}{200})^c$

$=0.0678925 \pi^c$

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