# Derivatives of Sine Fucntions

Math Portfolio Assignment:

Derivatives of Sine Functions

Bridget Belsher

Derivatives of Sine Functions, Type I

Date Set: January 14 2008

Date Submitted: February 5 2008

Father Lacombe High School

Mrs. Gabel

I certify this portfolio Assignment is entirely my own work

The above graph, displays the unaltered sin function, in its original form. The functions domain on the graph is -2π ≤ 2π. The result of this is can be seen in the two oscillations with the range. Its behavior can be determined by the behavior of the first oscillation (the oscillation starting at -2π), as the same behavior will be observed in the second oscillation (that starting at 0), due to the repetitive nature of sine functions. From the point (-2π, 0), the slope can be described as increasing with a value of one.  However, one can clearly see that at ( the function has no slope, as it is a point of discontinuity, that is having a slope of zero, as the line is straight. Just after that point, the slope is -1, which causes a constant decrease in the functions y-value and increase in the functions x value, which continues until the point (π, -1), at which it reaches another point of discontinuity.

The derivative on sin(x) is Cos(x), which is seen in the graph below and further explained as well.

Cos(x)’s behavior describes the slope of sine(x). This can be seen, if you compare the to graphs at the point  , where the slope of sin(x) is 0 and the Cos(x)’s actual value is 0, being (π/2,0) . The graph for the derivative of Sin(x) can also be found using a GDC, with no knowledge of the behavior of Cos/Sin functions prior to this paper. This can be done by going pressing the [y=] then going [2nd] [catalogue] [log] and then scrolling down to [nDeriv]. From here you press [enter] and plug in the necessary so that you end up with it looking something like y= nDeriv [Sin(x), x, x].  The resulting graph is displayed on the next page, and is labeled as f’(x). The seond graph displayed on the next page is labeled f(x) and is of the original function, f(x)=sin(x). The final graph on the next page displays both F’(x) and F(x).

The fact that f(x)=sin(x) derivative is indeed, f’(x)=cos(x) will be  proved in the equation on the next page, through the use of limits.

f(x) = sinx

Lim f(x + h) – f(x)

h→ 0            h

Lim sin(x + h) – sin(x)

h→ 0                h

Lim sinxcosh + sinhcosx – sin(x)

h→ 0                           h

Lim sinx(cosh – 1) + sinhcosx

h→ 0                           h

Lim sinx(cosh – 1)  +  Lim sinhcosx

h→ 0              h                 h→ 0      ...