Math Problem Statement
Graphs of Trigonometric Functions y = sin (3x) + cos (2x)
Solution
To analyze the graph of the function , let’s break down its components and then outline the key characteristics of the graph.
Step 1: Understand the Components
The function is a combination of two trigonometric functions with different frequencies:
- : The frequency here is , meaning it completes three cycles within a interval. The period for is .
- : The frequency here is , so it completes two cycles within a interval. The period for is .
Step 2: Determine the Behavior of
Since this function combines sine and cosine functions with different frequencies, its graph exhibits complex oscillations rather than simple sinusoidal behavior. Specifically:
- Periodicity: The graph repeats itself over the least common multiple of the two component periods ( and ), which is .
- Amplitude: The maximum amplitude of the graph will be influenced by the sum and difference of the amplitudes of and , which range between -1 and 1.
Step 3: Graph Characteristics
- Frequency Interference: The overlapping of the sine and cosine waves results in a waveform with varying amplitude over the interval.
- Symmetry: This function is neither purely even nor odd, as does not satisfy the typical symmetry properties of even or odd functions.
- Zeros and Peaks: The function will have multiple intersections with the x-axis (zeros) and variable peaks due to the phase differences between and .
Would you like to see a detailed plot of this function, or do you have further questions?
Here are some questions to explore:
- How does changing the coefficients of in and affect the period of the function?
- What is the amplitude range for the sum of two trigonometric functions with different frequencies?
- How do we find the exact points where ?
- What does the frequency interference look like graphically?
- What would happen if the function were instead?
Tip: When dealing with sums of sine and cosine functions, analyze each function’s period and amplitude separately before combining them. This helps in predicting the behavior of the overall function.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Trigonometry
Sine and Cosine Functions
Graphing of Trigonometric Functions
Periodicity
Amplitude
Formulas
y = sin(kx)
y = cos(kx)
Period of y = sin(kx) = 2π / |k|
Period of y = cos(kx) = 2π / |k|
Theorems
Principle of Superposition of Waves
Least Common Multiple of Periods
Suitable Grade Level
Grades 10-12