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Trigonometric Functions

Common to all trig functions are three properties.

  1. The amplitude of the trig function tells what the range is.
  2. The period tells you how often the function repeats itself.
  3. The phase tells you whether it is shifted to the right or left of its usual starting location.

The most common trig functions are the sine and cosine, written tex2html_wrap_inline430 and tex2html_wrap_inline432 respectively. The next most common function is the tangent, written tex2html_wrap_inline434 which is the quotient tex2html_wrap_inline436 . Sine and cosine both have a standard period of tex2html_wrap_inline438 while tangent has a period of tex2html_wrap_inline172 . Another difference is that tangent has a vertical asymptote wherever tex2html_wrap_inline442 , ie. at tex2html_wrap_inline182 .

The general form for the sine function and cosine function is

equation84

where A is the amplitude, tex2html_wrap_inline448 the period, and tex2html_wrap_inline450 the phase. Normally, sine and cosine oscillate between tex2html_wrap_inline452 , with an amplitude, they oscillate between tex2html_wrap_inline454 . The phase simply shifts the function to the right by tex2html_wrap_inline456 . Note that for these functions, the argument x is given in radians. The following properties are useful.

  1. tex2html_wrap_inline460
  2. tex2html_wrap_inline462
  3. tex2html_wrap_inline464
  4. tex2html_wrap_inline466

There are several other trig functions. Secant is the reciprocal of cosine, cosecant is the reciprocal of sine and cotangent is the reciprocal of tangent.

There are also some useful relations between the exponential functions and the trig functions. For example, Euler's formula says that

equation93

This can be rearranged to discover that

equation96

where tex2html_wrap_inline468 .



Vector Calculus
Wed Sep 17 14:50:13 MST 1997