Image Sine functions make good mathematical models for studying periodic or rotary motion because they repeat at regular intervals

Graph of Sine

Why do we need to graph the sine and cosine functions? The answer is that a graph can help us visualize properties of sine and cosine functions.

The graph of y = sinSymbol Theta can be drawn by plotting a number of points (Symbol Theta, sinSymbol Theta) as Symbol Theta takes a series of different values. Since the sine function is continuous, we can join these points with a smooth curve.

The following is the table of values for the interval [0, 2Symbol PI]:

degrees
radians
sinSymbol Theta 		0°
0.0
0.0 		30°
Symbol PI/6
0.5 		45°
Symbol PI/4
0.707 		60°
Symbol PI/3
0.866 		90°
Symbol PI/2
1.0 		120°
2Symbol PI/3
0.866 		135°
3Symbol PI/4
0.707 		150°
5Symbol PI/6
0.5 		180°
Symbol PI
0.0
degrees
radians
sinSymbol Theta 		180°
Symbol PI
0.0 		210°
7Symbol PI/6
-0.5 		225°
5Symbol PI/4
-0.707 		240°
4Symbol PI/3
-0.866 		270°
3Symbol PI/2
-1.0 		300°
5Symbol PI/3
-0.866 		315°
7Symbol PI/4
-0.707 		330°
11Symbol PI/6
-0.5 		360°
2Symbol PI
0.0

Image

As shown by the graph, the sine function is periodic with a period of 2Symbol PI; the graph repeats the shape on the interval [0, 2Symbol PI]. The pattern on the interval [0, 2Symbol PI] repeats itself indefinitely over intervals with a length of 2Symbol PI both to the left and to the right.

Please conduct the experiment to study the properties of sinusoidal graphs of the general form y = A sin B(Symbol Theta-C), in which

                A is the amplitude,
                B is the number of cycles the graph makes in 2Symbol PI (period = 2Symbol PI/B),
                C is the phase shift.

  [Experiment] [Exercise]