In the antiquity the
sine was used as the length of a chord. Given a circle with a certain radius, for a large
amount of values of the angles the lengths of chords were written down 1). Such cord tables can be found in
(150 AC) and in the Indian Surya Siddanta, and were used especially by
In the late Middle Ages there was a short period that one used to write the tangent as
sixtieths of the radius, and sixtieths of sixtieths. It was the German astronomer Regiomontanus
(Johannes Muller, c. 1450), who prevented this practice. He wrote
the trigonometric functions as lengths of chords for a circle with a radius of 10n
(n = 5 - 15), so that the results could be written as whole numbers.
It was not earlier than the 19th century that the sine got his actual meaning as the
proportion of line segments.
Geometrically the sine can be defined, in a rectangular triangle (Euler) as the proportion
of the opposite side to the hypotenuse. Another way to define the curve is as a power
A sine curve is also called a sinusoid.
A point moves with constant speed on the circumference of a circle. Take as center of the
circle the origin, then the coordinates of the moving point as function of the rolled
angle a are: (cos(a), sin(a)). So the component along an arbitrary diameter is a sine
function of time. This is the definition of a harmonic
oscillation. Necessary for this movement is a force, which is proportional to the
movement: F = - k x.
The sine obeys this rule, because for its second derivative applies: y'' = - y.
In fact, the harmonic oscillation is the form in which we meet the sine in our physical
A moving harmonic oscillation gives as result a sinusoid wave. The periodicity of the sine
is called the wavelength, the amplitude is the factor of sin x.
Imagine a triangle, the sine formula states that the ratio of the sine of an angle and the
opposite side is equal for all three angles.
An equalized sine is the result when the negative values have been
turned: y = |sin x|.
Sine means bend, curve, bosom. It is a literal translation of the Arab gaib, what has been
derived from gib, a way to spell the Indian jya (cord). So there is a relation
old habit of using the sine as the length of a cord. In natural life we see the sine form
in the bosom of a wife.
In screwing activities we make a three dimensional helical
line, when projecting the line on a surface through the screw's axis the
result is again a sine.
In the bicycle-loving country of Holland, research led to the conclusion that a
street threshold (to limit the speed of the cars) in the form of a sine is the
most comfortable form for bicycling.
The cosine is defined - in the same rectangular
triangle - as the proportion of the adjacent side to the hypotenuse. The cosine is a
translated sine: cos x = sin (x + π/2).
The cosine formula is an extension of the theorem of Pythagoras. For a triangle without a
rectangular angle, it states that a2 = b2 + c2 - 2bc cosφ3).
Derived from the sine are some functions, that have been used in older times:
- the versine or versed sine is defined as: versin(x) or vers(x) = 1 - cos(x)
- the vercosine or versed cosine is defined as: vercosin(x) = 1 + cos(x)
- the coversine or coversed sine is defined as: coversin(x) = 1 - sin(x)
- the covercosine or coversed cosine is defined as: covercosin(x) = 1 + sin(x)
- the haversine or haversed sine is defined as haversin(x) or hav(x) = 1/2 versin(x)
- the havercosine or haversed cosine is defined as havercosin(x) = 1/2 vercosin(x)
- the hacoversine or cohaversine or hacoversed sine is defined as hacoversin(x) = 1/2 coversin(x)
- the hacovercosine or cohavercosine or hacoversed cosine is defined as hacovercosin(x) = 1/2 covercosin(x)
The inverse functions of the sine and the cosine are
called the arc sine arcsin(x) and the arc cosine
arccos(x), respectively. These are cyclometric functions.
Sine series and cosine series are obtained while adding sines or
cosines, in the sine summation.
1) The length of a cord can be written in our
current notation as: cord(a) = 2 R sin (a/2)
2) The sine as a power series:
3) The angle π is the angle opposite line a.