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Understanding Trigonometric Functions: Cosine Example

Introduction to Trigonometric Functions

• The video begins with an introduction to trigonometric functions, specifically focusing on the cosine function and its graphical representation.

• It references a previous video that covered the basic parameters of the sine function, setting the stage for understanding cosine.

Graphing the Cosine Function

• The cosine function is graphically represented from 0 to 2pi, with a crucial period of 2pi. This is essential for understanding its behavior.

• The general model for cosine is presented as a cdot cos(bx + c) + d, where:

• a represents amplitude,

• b affects the period,

• c indicates phase shift,

• d denotes vertical shift.

Calculating Key Parameters

• To find important characteristics of the cosine function, formulas are introduced:

• Period (T): Calculated as T = 2pi/b. For this example, with b = 4, it simplifies to pi/2.

• Phase Shift: Determined by -c/b; in this case, it results in a positive shift due to double negatives. Thus, it becomes pi/8.

Constructing Data Points for Graphing

• A table is created to identify key coordinates necessary for accurately graphing the cosine function:

• Starting point at zero and progressing through standard angles like pi/2, pi, and so forth up to 2pi. These points help establish where the graph will peak or trough.

Finding Actual Points for Graphing

• The process involves dividing original x-values by b and adding/subtracting phase shifts:

• For instance, starting from zero gives a result of pi/8.

• Continuing this method yields additional points such as pi/4, 3pi/8, etc., which are critical for plotting accurate positions on the graph.

Finalizing Coordinates and Amplitude Representation

• After calculating all necessary coordinates based on established rules, these values are plotted against their respective amplitudes.

• The amplitude here is noted as being equal to two (from earlier calculations), indicating maximum heights reached by the wave form during oscillation between values of +2 and -2. This helps visualize how high or low the wave peaks relative to its centerline position.

Graphing Trigonometric Functions

Understanding the Graph of a Cosine Function

• The graph is constructed with key coordinates: starts at 1/8 , reaches maximum amplitude of 2, and crosses zero at pi/4 .

• The period of the function is confirmed to be pi/2 , calculated by subtracting the starting point from the endpoint.

• The calculation for the period shows that 5/8 - 1/8 = 4/8 = pi/2 , confirming accuracy in graphing.

Transition to Another Trigonometric Function

• Introduction of another trigonometric function, emphasizing its mathematical model as acos(bx + c) + d .

• Key parameters identified: amplitude (1), vertical shift (-1), and phase shift ( -pi/4 ).

Calculating Period and Shift Points

• The period is derived from dividing 2pi by b (which equals 2), resulting in a period of pi .

• The shift point is calculated using the formula for horizontal shifts, yielding a value of -pi/8 .

Determining Original Coordinates for Graphing

• Original cosine function coordinates are established: starts and ends at 1, with critical points at multiples of x = 0, pi/2, 3pi/2, ....

• A formula is used to find new coordinates based on original values adjusted by b and displacement.

Finding New Coordinates Efficiently

• For each original coordinate, adjustments are made using the relationship between original x-values and their corresponding shifts.

• Notably, differences between points are consistently found to be pi/4 , allowing for quick calculations across all coordinates.

Finalizing Graphical Representation

• Each new coordinate is determined by adding increments of pi/4 , ensuring consistency in periodicity.

• This method allows efficient plotting without recalculating every individual point.

Adjustments Based on Vertical Shifts

Function Graphing and Amplitude Analysis

Understanding Function Displacement and Segmentation

• The function's axis is positioned similarly to the x-axis, with a displacement along a drawn line. The segments are divided into eighths, indicating where specific values will be placed on the graph.

• Positive eighths are marked by duplicating segments; this results in coordinates being assigned to each segment, such as -1/8 for the first segment.

Analyzing Amplitude and Function Behavior

• The function has an amplitude of +1, meaning it will reach its maximum at 1 and minimum at -2. This indicates that the function oscillates between these two points due to its defined amplitude.

• A sinusoidal function typically starts at its peak (1), returning to this value after completing a cycle. In this case, it begins high and ends high, reflecting standard behavior for cosine functions.

Graphing the Function