Graphing Sine and Cosine Functions: Domain, Range, and Period

TLDR; Graphing sine and cosine functions on the coordinate plane, determining domain, range, and period, using the unit circle and tables to plot key values.

📈 Graphing Goals

The goals of the video are to graph the sine and cosine functions on the coordinate plane using the unit circle.

The video aims to determine the domain and range of the sine and cosine functions, as well as to establish the period of these functions.

The speaker emphasizes the importance of determining where the terminal side of an angle in standard position intersects the unit circle, linking the x coordinate of that point to cosine theta and the y coordinate to sine theta.

🔄 Wolfram Demonstration

A Wolfram demonstration is used to animate a point around the unit circle and observe the graph of the sine function by plotting y-coordinates of the point.

The demonstration illustrates the cycle of the sine function, showing that the period equals 2pi radians, and also covers the graph of the sine function for negative angles.

The speaker highlights the usefulness of looking at the decimal values of the function at different angles, filling out a table from the calculator to plot key values of sine theta.

📊 Plotting Sine Function

Using the table created, the speaker demonstrates how to plot key values of the sine function on the coordinate plane, dividing the angles into fourths and showing the corresponding values of sine theta.

The resulting graph of the sine function is explained based on the plotted values, and the domain and range of the function are discussed, highlighting that the range will always be between -1 and +1.

📉 Graphing Cosine Function

The speaker uses a Wolfram demonstration to show the graph of the cosine function by focusing on the x coordinates of the point around the unit circle.

The process of filling out a table to plot key values of the cosine function is explained, and the resulting graph is presented, emphasizing that the domain is all reals and the range is from -1 to +1.

The periodicity of the cosine function is discussed, with the period being 2pi radians and the amplitude also being 1.

🔄 Periodic Functions

The concept of periodic functions is introduced, defining a function as periodic if f of x + p = f of x for all values of x.

The constant 'p' is identified as the period, required to be positive, and it is explained that a function with period 'p' will repeat on intervals of length 'p'.

The properties of the sine theta and cosine theta graphs are summarized, including their continuity, period, x-intercepts, and symmetrical properties.

🎢 Graph Summary

The speaker provides a summary of the properties of the sine theta and cosine theta graphs, including their continuity, period, x-intercepts, and symmetrical properties.

The video closes with a statement of gratitude for watching and a note of thanks to the viewers.