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 and respectively. The next most common function is the tangent, written which is the quotient . Sine and cosine both have a standard period of while tangent has a period of . Another difference is that tangent has a vertical asymptote wherever , ie. at .

The general form for the sine function and cosine function is

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

1. 
2. 
3. 
4. 

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

This can be rearranged to discover that

where .