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Angles – The seed of Trigonometry

Posted in Mathematics, Methinks by ytelotus on June 1, 2013

Angle in degrees or radians? Which one to choose?

Why do we express angles in terms of fractions or multiples of π i.e. 30°, 90°, 180°, 360°… as  π/6, π/2, π, 2π … ?

Why does a circle have 360°?  Why is 360° = 2π?

What is behind the periodicity of trigonometric functions?

Introduction

An angle is a measure of rotation of a ray OA about its initial point.

The concept of angle and its units of measurement are best understood with a little knowledge of the contexts in which angle took its very birth and flourished thereon :

<Phase I : The birth of Angle via Line geometry>

Angle is a concept that evolved out of Line geometry i.e. geometry involving the meeting of straight lines.  This makes it an integral attribute of polygons, where lines join to make a closed circuit.  The ancient people who studied triangles in particular, observed a relation between the  angles and the ratios of the sides of a right-angled triangle :

For the two similar triangles shown in Fig 2, it is seen that, for a given angle Θ,

X/Z = X`/Z`

Y/Z = Y`/Z`

X/Y = X`/Y`

The ratios being functions of Θ were subsequently tabulated for varying values of Θ and came to be known as sin(Θ), cos(Θ) and tan(Θ) respectively.  This field of study of the relationship between the angles of triangles and the ratios of its sides came to be known as Trigonometry and the functions – Trigonometric functions.  It helped mankind a great deal in the measurement of distances and heights of remote inaccessible objects scaling up to astronomical lengths.

<Phase II : The growth of Angle via Circular geometry>

Angle is a natural attribute of a Circle.  When the rotation of a ray OA about its initial point O is complete, the locus of the path is a circle.  At any instant of revolution along the circular path, the projections of A onto the x and y axes give us a rectangle with the diagonal OA`bisecting the rectangle into two right-angled triangles.  Thus the unfolding of angle Θ along a circle can be mapped to right-angled triangles varying along with Θ, linking trigonometric functions with circular geometry.

The beauty of linking an angle with circle is that, as A moves along the circumference revolutions after revolutions, Θ keeps growing with infinity being its limit.  This treatment extends the span of trigonometric functions to infinity.  Because a circular movement is repetitive in nature, the graph of trigonometric functions should also repeat for every revolution of Θ.  It naturally follows that trigonometric functions have been used to model the periodic phenomena in nature, such as the motion of waves, simple harmonic motion, sound vibrations etc.

Angular measurement

Two popular units of measurement of angles have emerged over time.  They are (1) Degrees and  (2) Radians.

(1) Degrees

The degree is a very ancient unit of measurement, so much so that its origin is not exactly established yet.  Multiple theories have been posited by those who sought to solve the mystery of  360 degrees associated with circles.  These are the two that stand out :

– It is believed that the number 360 is an approximation of the number of days in a year.  It is said that the ancient astronomers, who were keen observers of the sky, used the delta shift in the daily positions of the stars and constellations as a reference to mark a degree.

– Another theory suggests that the basis of degrees, minutes and seconds is the sexagesimal system.  Sexagesimal is a numeral system with sixty as a base.  The number 60 being a highly composite number i.e. it has twelve factors {1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60}, was perhaps adopted by the Sumerians and the Babylonians for the convenience of measurements.

Both the theories can be equally argued in favor of.  The fact that a year has 365 ¼ days and a circle has 360 degrees is quite a coincidence.  However, if we were to presume that there was a numeral system in vogue, at the time degree was fixed to be the unit of angular measurement, then this theory of sky observation does not appeal.  The fact that a degree is further divided into 60 minutes and a minute is sub-divided into 60 seconds is an evidence of the permeance of sexagesimal system into the angular measurement.

That evokes a question :

If degree is a unit based on sexagesimal system, then why doesn’t a circle have 60 degrees instead of 360 degrees?

This perhaps has to do with the chronology of geometry.  The concept of angles emerged out of the study of lines and triangles first.  Further, an equilateral triangle makes the same angle at its three corners.  That must have made it convenient to use it as the basic unit of angle measurement, by marking each angle to be of 60 parts based on the sexagesimal system.

Over time, when the concept of angles extended to circles, it was convenient to use equilateral triangle with side length equal to the radius of the circle as shown in Fig. 5, to measure the angle enclosed by circles.

Placing six such triangles together gave a circle with 360 degrees angle.

The radian marks its birth around the 18th century on the timeline.  A radian is defined as the angle subtended at the center of a circle by an arc equal to the radius of the circle.

A visual comparison of Fig. 7 with Fig. 5 says much in one glance.  Fig. 7 carves out a sector of arc length equal to the radius enclosed by two radii.  Fig. 5 carves out an equilateral triangle with radii as its sides.  (Yes, the equilateral triangle is further divided into 60 parts, whereas the sector is not.  That explains the finer granularity of degree measure.)

The radian measure is based on a unique property observed of circles : For a circle, the ratio of the circumference to the diameter (= twice the radius) is a constant, regardless of its size.  This constant called ‘pi’ (written as Π) is an irrational number that approximates to a value of 3.14159.

When we calibrate angle in terms of radians, we actually fragment the circumference into arcs of length equal to the radius.  Fig. 8 shows six complete arcs with one fractional part making it about 6.28318 radian measure i.e. 2Π radians makes one revolution.

An angle specified in radians is therefore a direct measure of how many times the radius occurs on the circumference of a circle for that angle.  For instance, three radians provide an arc coverage equal to three radii length.  This means, if the circle rotates by a measure of three radii length, it implies an angle coverage of three radians.  This maps the math of radians to the physics of rotatory movement also viewed as periodic movement.

Radian is a dimensionless entity – a ratio of the arc length covered by the angle to the radius of the circle.  Giving it the name of radian makes it convenient for use.