Sinusoid

above: “Fiery”. Plot of the equation Sin[x* Sin[y]] == Cos[y*Cos[x]]. fiery.gcf

Mathematica Notebook for This Page.

History

All trignometric functions sine, cosine, tangent, secant, cosecant, cotangent can all be simply defined in terms of a single function sine. Sine, as associated with trignometry, began in early civilization as a very important measuring science. When the function concept and calculus and analytic geometry were introduced in about 1700, sine became a function and has little to do with triangles. The sine function appears unexpectedly throughout analysis, because in essence it captures the idea of a wave, a fundamental concept in physics.

Excerpt from Robert C Yates (1974):

«Trigonometry seems to have been developed, with certain traces of Indian influence, first by the Arabs about 800 as a aid to the solution of astronomical problems. From them the knowledge probably passed to the Greeks. Johann MÜller (c.1464) wrote the first treatise: De triangulis omnimodis; this was followed closely by others.»

See also: History of trigonometric functions↗.

Description

Sinusoid is the curve of the sine function. Sine is sometimes called circular function because the essential feature of the sine function can be thought of as a point moving around a circle in a uniform way, and the value of sine being the height of the point.

Step by step description:

1. Let A be a point on origin.
2. Let C be a point on the positive x-axes.
3. Let D be a point on (-1,0).
4. The sine function at Distance[A,C] is the height of E, where E is a point such that ArcLength[A,D,E] == Distance[A,C].

Sine Curve

Formulas

In the formula y == a*Sin[x/p+s], a is the amplitude, p the period, and s the phase shift. sine_plot.gcf

Trig Functions In Terms of Sine

All trig functions is defined in terms of sine.

If a right triangle is placed in a stardard position (That is: in the Cartesian coordinate system such that it lies in the first quadrant, and the right angle vertex lies on the x-axes, and the hypotenuse touches the origin), and if r denote (the length of) the hypotenuse, x the bottom side, y the verticle side, θ the angle of x and r, then we have the following formulas:

Trig Identities

See: List of trigonometric identities↗.

Properties

Basic Trig Functions

above: The blue is sine curve. The pink is cosine. The red is tangent. ( trig.gcf)

above: The blue is sine curve. The pink is cosecant, the red is cotangent. One can clearly see that sin and csc are multiplicative inverses: the smaller the value of sine, the larger is cosecant, and vice versa. ( trig2.gcf)

Helix Projection

Sinusoid is the orthogonal projection of the space curve helix. (see helicoid) A helicoid is a surface formed as the trace of a rotating a line along a axis.

development of cut cylinder

Sinusoid is the development of a obliquely cut right circular cylinder. (the edge of the cylinder rolled out is a sinusoid). graphics code..

Wavy Surface

above: A packaging form is modeled after the surface Sin[x]*Sin[y]. graphics code.

above: the surface Sin[x*y]+Sin[y*z]+Sin[z*x]==0 borg_cube.gcf.

Related Web Sites

Robert Yates: Curves and Their Properties.

Wikipedia: sinusoid↗.

Wikipedia: Trigonometric functions↗.

© 1995-2008 by Xah Lee.