Math Problem Statement

Graphs of Trigonometric Functions y = sin (3x) + cos (2x)

Solution

To analyze the graph of the function y=sin(3x)+cos(2x)y = \sin(3x) + \cos(2x), 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:

  1. y=sin(3x)y = \sin(3x): The frequency here is 32π\frac{3}{2\pi}, meaning it completes three cycles within a 2π2\pi interval. The period for sin(3x)\sin(3x) is 2π3\frac{2\pi}{3}.
  2. y=cos(2x)y = \cos(2x): The frequency here is 22π\frac{2}{2\pi}, so it completes two cycles within a 2π2\pi interval. The period for cos(2x)\cos(2x) is π\pi.

Step 2: Determine the Behavior of y=sin(3x)+cos(2x)y = \sin(3x) + \cos(2x)

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 (2π3\frac{2\pi}{3} and π\pi), which is 2π2\pi.
  • Amplitude: The maximum amplitude of the graph will be influenced by the sum and difference of the amplitudes of sin(3x)\sin(3x) and cos(2x)\cos(2x), which range between -1 and 1.

Step 3: Graph Characteristics

  1. Frequency Interference: The overlapping of the sine and cosine waves results in a waveform with varying amplitude over the 2π2\pi interval.
  2. Symmetry: This function is neither purely even nor odd, as y=sin(3x)+cos(2x)y = \sin(3x) + \cos(2x) does not satisfy the typical symmetry properties of even or odd functions.
  3. Zeros and Peaks: The function will have multiple intersections with the x-axis (zeros) and variable peaks due to the phase differences between sin(3x)\sin(3x) and cos(2x)\cos(2x).

Would you like to see a detailed plot of this function, or do you have further questions?


Here are some questions to explore:

  1. How does changing the coefficients of xx in sin\sin and cos\cos affect the period of the function?
  2. What is the amplitude range for the sum of two trigonometric functions with different frequencies?
  3. How do we find the exact points where y=sin(3x)+cos(2x)=0y = \sin(3x) + \cos(2x) = 0?
  4. What does the frequency interference look like graphically?
  5. What would happen if the function were y=sin(3x)+sin(2x)y = \sin(3x) + \sin(2x) 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.

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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