Thursday, October 27, 2011

Trigonometry

When we talks about trigonometry, we will most probably think of trigonometric functions, e.g. sine, cosine, tangent etc. Ever think of how were the graphs of these functions obtained?

Let's consider a right triangle with hypotenus (r) of 1 unit as in the graph below (left column). If you tick the first function  at the right column, you will see the appearance of point E with the coordinate in the right column. Take note that  in the coordinate is shown in radian mode.

Recall the definition of the trigonometry functions as below, observe the changes of value f  (the height of point E) as you  slowly drag the point C in the graph in the first column and observe the changes of the angle.





As  at the left column increases in the first quadrant, point E is getting higher because y-value in (x,y) is getting bigger (the triangle is getting taller), resulting  also increases; while the hypotenus, r remains as 1.


Now, if you continue to explore other functions like and , point F and point G will appear. Take note that Point E is having the coordinate of , point F is and point G is . Observe the position of point E, F or G as you drag point C. You may choose the desire graph at the bottom right corner of the graph.

If you wish to know more about trigonometric functions, please click on the equation you are interested under Trigonometry at the right column.

Betty, Created with GeoGebra

Friday, October 21, 2011

Hyperbolic Tangent & Hyperbolic Cotangent

Hyperbolic tangent is defined as
,

while hyperbolic cotangent is the reciprocal of hyperbolic tangent, and it is defined as
.

If we want to sketch the graph of the earlier function, think of the y-intercept and limits below:
  1. As , ;
  2. when x = 0, ;
  3. as , .

If we want to sketch the graph of the later function, think of the limits below:
  1. As , ;
  2. as , ;
  3. as , ;
  4. as , .

The concept above also applied to and for all values of a > 0. You may slide the value of 'a' below to look at the changes of and for different values of 'a'.


Betty, Created with GeoGebra

Hyperbolic Cosine & Hyperbolic Secant

The hyperbolic cosine function is defined as

or generally,
.


Look at the graphs below, the blue solid curve is the curve of  , whereas the dotted curves are the graphs of its terms. If you can't remember the graph of , you can try to add up the y-values of the two terms (dotted curves) and get the y-value of .

Try to slide the value of a to different values to view the changes of the graphs.



Now, take a look at the graph of the reciprocal of hyperbolic cosine function, i.e. hyperbolic secant function, which is defined as

Observe that when a = 1,
  1. as , , thus ;
  2. as , , thus;
  3. as , , thus .

Again, slide the value of a to view the changes of the graphs.


Betty, Created with GeoGebra

Hyperbolic Sine & Hyperbolic Cosecant

The hyperbolic sine function is defined as


or generally,


Look at the graphs below, the blue solid curve is the curve of  , whereas the dotted curves are the graphs of its terms. If you can't remember the graph of , you can try to add up the y-values of the two terms (dotted curves) and get the y-value of .

Look at the graph below, try to slide the value of a to different values to view the changes on graph.





Now, take a look at the graph of the reciprocal of hyperbolic sine function, i.e. hyperbolic cosecant function, which is defined as
.

Observe that when a =1,
  1. , , thus ;
  2. , , thus ;
  3. , , thus ;
  4. , , thus .
Again, slide the value of a to view the changes of the graphs.

Betty, Created with GeoGebra