Saturday, November 9, 2013

Why An Angle Of 180° Can Be Represented By Pi (π) ?

Why An Angle Of 180° Can Be Represented By π ?

by cawan (cawan[at] or chuiyewleong[at]

on 09/11/2013

An angle can be represented in two units, degree (°) or radian (rad).  In the unit of degree, a full circle is 360°.  So, half circle is 180° and quarter circle is 90° (orthogonal).  However, in the unit of radian, 1 rad is defined as the arc of a circle has same length as the radius (j) of that circle.  We all know the perimeter of a full circle is 2πj.  So, a full circle is 2πj / j = 2π rad, a half circle 2π / 2 = π rad, and of course a quarter circle is π/2 rad.  Anyway, both of them are just a representation of an angle, nothing special.  When a an angle becomes a parameter of a trigonometry function such as cosine, then it makes more senses.  We know the cosine of an angle is the ratio of adjacent to the hypotenuse.  So, when an angle is near to 0 rad, the adjacent is almost equal to hypotenuse.  Thus, the cosine of 0 rad is 1.  While the angle start to increase, the adjacent will decrease accordingly.  When the angle is near to π/2 rad or 90° (quarter circle), the adjacent will close to 0.  Hence, the cosine of π/2 rad or 90° is 0.  If the angle keep increasing, the adjacent will start to increase again, but in opposite direction.  When the angle is near to π rad or 180° (half circle), the adjacent becomes almost equal to hypotenuse again.  So, the cosine of π rad or 180° is -1 (remember the adjacent is in opposite direction ?).  For the same reason, the cosine of 270° and 360° are 0 and 1, respectively.  If we plot the variation of cosine of an angle from 0° to 360° and keep repeating, then we get a standard cosine wave.  The variation of angle from 0° to 360° is defined as a cycle.  The variation speed of the angle is defined as angular frequency (ω) in the unit of radian per second, which concerning how many cycle can be completed in 1 second.  For this reason, a frequency in the unit of Hz can be expressed in radian per second by multiplying with 2π (remember 2π rad is equal to 360° ?).  Thus, ω = 2πf.  In addition, a cosine wave can be expressed in cos(ωt), where the t is the time axis.  Now, how about the sine of an angle ? Well, I will leave this as your own exercise.

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