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Home < Volume 2 Three Skills For Algebra << Chapter 20. Degrees and Radians
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Degrees and RadiansVolume 2, Three Skills for Algebra 1 Lengths of Circular Arcs
The length $s$ of a circular arc is proportional to the angle $\theta$ its spans. \] Geometrically this proportionality means if the angle $\theta$ is double or tripled, so is the arclength $s.$ Mathematically, this relationship is described by the relation \[s=k\theta\] where $k$ is called the constant of proportionality. Graphing $s$ versus $\theta$ where $ 0^\circ \le \theta \le 360^\circ,$ gives a get a straight line segment with slope $m=k.$ The assumption that the circumference of the circle is $2\pi r$ implies $2\pi r = k\cdot 360^\circ$ where $r$ is the radius. This in turn implies \[k =\frac{2\pi r}{360^\circ} = \frac{\pi r}{180^\circ} \def\degree{\hbox{ degree}} \def\radian{\hbox{ radian}} \def\revolution{\hbox{ revolution}} \] The unit of angle measurement is the degree $1^\circ =1 \degree.$ 2 Angles and ArclengthThe foregoing implies that \[s=k\theta =\frac{\pi r}{180^\circ} \cdot \theta =\frac{\pi}{180^\circ}r\theta\] This in turn implies \[\frac sr = \frac{\pi}{180^\circ}\theta\] or equivalently that \[\theta=\frac{180^\circ}{\pi}\frac sr.\] Thus the real number $\frac sr$ is proportional to the angle $\theta,$ and vice-versa. Thus the ratio $\frac sr$ determines the angle $\theta,$ and vice-versa. So specifying one, specifies the other. Radians and Radian MeasureThe real number $\frac sr$ equals the number of times the radius $r$ goes into the arc length $s.$ This ratio, the real number $a=\frac sr$ is called the radian measure of the angle $\theta.$ Now let the new unit, namely the radian: \[ 1 \radian=\frac{180^\circ}{\pi}={57.2957795}\degree s\] Then the angle \[\theta=\frac{180^\circ}{\pi}\frac sr\] is also given by the expression $\theta = a \radian$ where as before $a=\frac sr$ is its radian measure. From \[1 \radian=\frac{180^\circ}{\pi}\] observe that \[1^\circ= 1.0 \degree =\frac{\pi}{180}\radian\] Note that the unit \[ 1 \revolution = 360^\circ = 360 {\degree}s = 2 \pi \radian \] The foregoing implies that \[s=k\theta =\frac{\pi r}{180^\circ} \cdot \theta =\frac{\pi}{180^\circ}r\theta\] This in turn implies \[\frac sr = \frac{\pi}{180^\circ}\theta\] or equivalently that \[\theta=\frac{180^\circ}{\pi}\frac sr\] Thus the real number $\frac sr$ is proportional to the angle $\theta,$ and vice-versa. Thus the ratio $\frac sr$ determines the angle $\theta,$ and vice-versa. So specifying one, specifies the other. Whence one may used in place of another. A like situation occurs in the choice of measure for length. The number of unit centimeters in a length, one measure of the length, is proportional to the number of units of meters in the length, another measure of the length. So measurements of equal value may be given in terms of either unit, centimeters or meters. Likewise, money may be counted in pennies and larger units: dollars, pounds, euros, yen and yuan. Radians and Radian MeasureThe real number $\frac sr$ equals the number of times the radius $r$ goes into the arc length $s.$ This ratio, the real number $a=\frac sr$ is called the radian measure of the angle $\theta.$ Now let the new unit, namely the radian: \[ 1 \radian=\frac{180^\circ}{\pi}=57.2957795^\circ\] Then the angle $\theta=\frac{180^\circ}{\pi}\frac sr$ is also given by the expression $\theta = a \radian$ where as before $a=\frac sr$ is its radian measure. From \[1 \radian=\frac{180^\circ}{\pi}\] observe that \[1^\circ= 1.0 \degree =\frac{\pi}{180}\radian\] Note that the unit \[ 1 \revolution = 360^\circ = 360 {\degree}s = 2 \pi \radian \] 4. Numerical Values of AnglesThere is a convention which identify radians, degrees and revolutions with real numbers. The conventions are as follows: \begin{eqnarray*} 1 \radian & = & 1.0 \\ 1 \revolution & =& 2\pi \radian \\ 1 \degree &=&\frac{\pi}{180}\radian \end{eqnarray*} The latter implies \[1^\circ=1 \degree=\frac{\pi}{180} \approx 0.0174532925 \] Note that for the sake of precision in algebraic reasoning, the ratio $\frac{\pi}{180^\circ}$ and its reciprocal $\frac{180^\circ}{\pi}$ should carried through calculations and only replaced by their approximations when actual computations are required. The earlier replacement is imprecise and doing it may cause opportunities for cancellation of the terms $\pi$ and $180^\circ$ to be missed. Angle Measurement RevisitedThe following diagram shows two concentric circles, and two arcs of lengths $s_1$ and $s_2,$ respectively, one on each.
To measure the angle of the inner arc in degrees, divide he perimeter of the inner circle into 360 equal-length arcs. Then count how many of these inner circle arcs are covered by the inner arc. The length of the inner arc is proportional to the second count. Both counts give the length of the outer and inner arcs in terms of one three hundred and sixtieth, that is $\frac1{360}$-th of the respective perimeters of each their circles In the above diagram, the two arcs, more precisely, arc lengths, s1 and s2 cover the same proportion of the circles of their respective circles. Therefore the two counts must be equal. One or both circles may represent circles could be the perimeter of a protractor. |
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Home < Volume 2 Three Skills For Algebra << Chapter 20. Degrees and Radians
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