Effect of Changing 'a' in aSin(nx) or aCos(nx) (Leaving Cert Mathematics): Revision Notes

Effect of Changing 'a' in aSin(nx) or aCos(nx)

Overview

The parameter aa in the functions $a \sin(nx)$ or $a \cos(nx)$ affects the amplitude of the trigonometric wave. Amplitude is the maximum vertical displacement of the wave from its equilibrium position $(y = 0)$.

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Key Observations

Role of $a$:

• $a$ determines the vertical stretch or compression of the sine and cosine graphs.
• The absolute value of $a$ $(|a|)$ gives the amplitude of the wave:

$$\text{Amplitude} = |a|$$

Effect of $|a| > 1$:

• The graph stretches vertically, making the wave taller.

Effect of $0 < |a| < 1$:

• The graph compresses vertically, making the wave shorter.

Effect of $a < 0$:

• The graph flips vertically, reflecting it across the $x$-axis.

Period and Shape:

• The period and shape of the wave are not affected by $a$; these are determined by $n$

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Worked Examples

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Summary

• Amplitude: |a| determines the maximum height of the wave from its equilibrium position.
• Stretch or Compression:
  • $|a| > 1$: Vertical stretch.
  • $0 < |a| < 1$: Vertical compression.
• Reflection: A negative $a$ flips the graph across the $x$-axis.
• The period and horizontal structure of the wave remain unaffected by $a$ Understanding the role of $a$ helps in graphing and interpreting trigonometric functions effectively.