Question 1079785
 

note: general formulas

    {{{y = a*sin(b*x)}}}
    {{{y = a* cos(b*x)}}}

The variable {{{b}}} in both of the following graph types affects the period (or wavelength) of the graph.

The {{{period}}} is the distance (or time) that it takes for the sine or cosine curve to begin repeating again. 

Frequency is defined as {{{frequency=1/period}}}.

The relationship between {{{b}}} and the period is given by:
{{{period =2pi/b}}}. Note: As {{{b}}} gets larger, the period {{{decreases}}}.

The variable {{{b}}} gives the number of cycles between  {{{0}}} and {{{2pi}}} . Higher  {{{b}}} gives higher frequency (and lower period).


Tip 1: The number {{{b}}} tells us the {{{number}}} of {{{cycles}}} in each {{{2pi}}}.

example:
For {{{y = 10cos(t), there is {{{one}}} cycle between {{{0}}} and {{{2pi}}}(because {{{b = 1}}}).

For {{{y = 10cos(3x)}}}, there are {{{3}}} cycles between {{{0}}} and {{{2pi}}} (because {{{b = 3}}}).

Tip 2: 
Remember, we are now operating using RADIANS. 
Recall that:{{{2pi= 6.283185}}} and that {{{2pi= 360}}}°



Now let's look at the graph of {{{y = 5sin( t)}}}.

This time we have  {{{a= 5}}} and {{{b=1}}}
{{{a= 5}}} is amplitude, so the curve goes up to 5 units and down to -5 units on the y-axis
 
since {{{b=1}}}, the period is {{{2pi/1=2pi}}} (periodic in {{{t}}} with period {{{2pi}}})


here is your sketch:

http://www.intmath.com/trigonometric-graphs/svg/svgphp-graphs-sine-cosine-amplitude-1-s1.svg