Recording-2025-03-06T13:27:59.061Z

Introduction to Sine and Cosine Functions

• Sine and cosine functions can be expressed in the form:K \times x

• This indicates that multiplying a constant (K) by x affects the function’s output.

• When looking at the expression K \times B, it produces another constant which also contributes to the function.

• Understanding how numbers interact with sine functions is crucial, as each number has a physical meaning.

Key Variables and Definitions

• Variables Introduced:

  • a: A constant term outside of the function.

  • k: The coefficient multiplied by x inside the function.

  • b: A constant that indicates shifts in the function.

• Amplitude:

  • The amplitude is the absolute value of a and represents the height of the sine function from the midline (y=0).

  • It relates to real-world phenomena like sound waves and signals.

Wave Characteristics

• Sine and cosine functions are wavy, suitable for modeling waves and signals (e.g., radio signals).

• The period:

  • Defined as the distance over which the function's cycle repeats.

  • It can be calculated irrespective of whether peaks or troughs are chosen, as long as the distance of repetition is considered.

The Role of K in Periodicity

• The value of k affects the period of the function:

  • Period Calculation:

    • The formula for period involving k is: ( \text{Period} = \frac{2\pi}{k} )

  • If k=1, it reflects the standard parent sine function without any stretching.

• Phase Shift:

  • Phase shift involves a horizontal shift in the sine or cosine graph and is evident when values are added or subtracted to x.

  • Mathematically represented in expressions like sine(Kx - B).

Analyzing Complex Functions

• In practices, sine functions expressed in forms like:

  • ( 3 \sin(2x - \frac{\pi}{2}) ) need to be manipulated to fit the standard forms.

• To reformat:

  • Factor a number (like 2) out of the x variable, considering transformations.

  • Adjust constants inside the function while maintaining equalities.

Period, Amplitude, and Phase Shift Review

• Identify known components in a complicated sine expression:

  • Amplitude: For example, with a coefficient of 3, the amplitude is also 3.

  • k value: This example shows k=2, leading to a period: ( \text{Period} = \frac{2\pi}{2} = \pi )

  • Phase Shift: For forms such as (x - \frac{\pi}{4}), after factoring, B is found to be (\frac{\pi}{4}) leading to a phase shift equation.

Plotting Sine Functions

• To plot:

  • Determine an easy point (for instance, the function value at x=0).

  • Identify y-axis crossing points (x-intercepts).

• Performance of each point under transformation helps facilitate plotting, such as:

  1. Zero Point: At x=0, y=0.

  2. Using Periodicity: Recognizes that the period oscillates across the y=0 baseline.

X Intercepts and Symmetry

• Investigate x-intercepts, which provide multiple points where the function intersects the horizontal axis.

• These x-intercepts can include values like (0, \pi, 2\pi), offering a basis for repetition and symmetry in sine curves.

• Thus, graphing becomes simpler knowing these properties.

Conclusion

• Utilizing amplitude, period, and phase shift greatly simplifies the graphing of sine functions.

• Understanding how to form standard sine equations allows quicker and accurate plotting rather than evaluating point-by-point.