Establishing Trignometric Identities

Establishing Trig Identities: hard for some people and fun for some people.

If you feel that establishing trig identities is hard then you need to change your perspective. Think of them as puzzle.

Example 1

Law of Sines: how it has been derived.

It can be observed that:

Therefore

and

Doing the same thing with the line drawn between angle A and side a will yield:

How to Graph Sine Function?

Today I want to talk about how to graph Sine functions. Graphing Sine is really easy. Now parent function of Sine is:

This is graph of f(x) = sin(x)

Function Sine is usually in y = a sin(bx ± c) ± d form.

• In that form "a" determines amplitude of the function. Amplitude is also known as height. Norman function has height of 1.
• Period:
  

• "b" determines period of the function.
• Formula of determining the period (p) of the function is ^2∏/_b.
• "
  ±^c/_b" is phase shift of the fuction.
  

Example 1 Graph the following function.

sin(2x)

In order to graph this function, you need to know three things.

1. Period - period of this function is = ^2∏/₂ =
   ^∏
2. Amplitude (a) = 1
   
3. Phase shift = 0

Graph:





Here is easy way to graph this function.

You know that the period is ∏ . You also know that sin(0) = 0.
You will start graphing from origin (0,0).

1. In the graph paper shown above, first blocks from 0 to ∏ . There are 6 blocks. Means that your period is 6 blocks long.
   
2. Second you need to divide 6 blocks in 4 equal blocks. So there will be 1.5 blocks.
   
3. At the starting of your first block (0,0) you will have y = 0. After 1.5 blocks you will have y = 1. After another 1.5 blocks you will have y = 0. Another 1.5 blocks you will have y = 1. At the end of the fourth block, you will have y = 0.
4. You also need to remember that in the Sine function, Zeros occur at every half period (3 blocks in this Example) and you max and means occurs at every period (6 blocks in this Example).

To graph Sine function (sin(2x)) by technique described above, first determine you period then count how blocks each period is. Then divide them in to four equal sized blocks then plot points (0, 1, 0, 1, 0). This is your one period. To graph it in whole graph paper, plot minimum "y" values and maximum "y" values every period and plot Zeros at every half period.