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

Cotangent Function

Cotangent function is a reciprocal for tangent function. If , then . Generally, cotangent function can be written as .

Now, consider only ,
as tan(x) approaches to 0, cot(x) approaches ∞ or -∞,
as tan(x) approaches ∞ or -∞, cot(x) approaches 0.

You may change the values of a = 1, b = 1, c = 0 and d = 0 and think about the construction of cot(x) from tan(x). After that, you may alter the values to look at the shifting, scaling or reflecting of the graphs.


Slide the values of a, b, c and d to observe the changes of the graph . What can you say about the effects of these values towards the function cot (x)?

Betty, Created with GeoGebra

Secant Function

Secant function is a reciprocal for cosine function. If , then . Generally, secant function can be written as .

Now, consider only ,

1. as cos(x) approaches to 0, sec(x) approaches to ∞ or -∞,
2. when cos(x) = 1, sec(x) = 1,
3. when cos(x) = -1, sec(x) = -1.

You may change the values of a = 1, b = 1, c = 0 and d = 0 and think about the construction of sec(x) from cos(x). After that , you may alter the values to look at the shifting, scaling or reflecting of the graphs.



Slide the values of a, b, c and d to observe the changes of the graph . What can you say about the effects of these values towards the function sec (x)?


Betty, Created with GeoGebra

Cosecant Function

Cosecant function is a reciprocal of sine function. If , then . Generally, cosecant function can be written as .

Now, consider only ,

1. as sin(x) approaches to 0, csc(x) approaches to ∞ or -∞,
2. when sin(x) = 1, csc(x) = 1,
3. when sin(x) = -1, csc(x) = -1.

You may change the values of a = 1, b = 1, c = 0 and d = 0 and think about the construction of csc(x) from sin(x). After that , you may alter the values to look at the shifting, scaling or reflecting of the graph.



Slide the values of a, b, c and d to observe the changes of the graph . What can you say about the effects of these values towards the function csc (x)?


Betty, Created with GeoGebra

Tangent Function

A tangent function generally can be written as f (x) = a tan (bx + c) + d.


Slide the values of a, b, c, and d to observe the changes of graph f (x) = a tan( bx + c ) + d. What can you say about the effects of these values towards the function tan (x)?

Betty, Created with GeoGebra

Cosine Function

A cosine function generally can be written as f (x) = a cos (bx + c) + d. The cycle repeated after every units. The amplitude of the curve is a units.


Slide the values of a, b, c and d to observe the changes of graph f (x) = a cos (bx + c) + d. What can you say about the effects of these values towards the function cos (x)?


Betty, Created with GeoGebra

Sine Function

A sine function generally can be written as f (x) = a sin (bx + c) + d. Try to move point A in the graph below, point B is the recurring point after one cycle. The period of the cycle is units and the amplitude is a units.


Slide the values of a, b, c and d to observe the changes of graph f (x) = a sin (bx + c) + d. What can you say about the effects of these values towards the function sin (x)?

Betty, Created with GeoGebra