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

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

Overview

The sine and cosine functions are periodic and oscillatory, defined for all real numbers. Their graphs exhibit repeating patterns, making them essential for modelling waves, circular motion, and other cyclic phenomena.

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Key Characteristics of Sine and Cosine Graphs

Amplitude

• Represents the maximum or minimum value of the function.
• Default amplitude: 1 (for $\sin x$ and $\cos x$)

Period

• The length of one complete cycle of the graph.
• Default period: $2\pi$

Key Points

• Sine: Passes through the origin and has peaks at $\frac{\pi}{2}$ and troughs at $\frac{3\pi}{2}$
• Cosine: Starts at its maximum value (1) and has peaks and troughs shifted $\frac{\pi}{2}$ from sine.

Symmetry

• Sine: Odd function $(\sin(-x) = -\sin(x))$
• Cosine: Even function $(\cos(-x) = \cos(x))$

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Graph Features

Sine Graph ($y = \sin x$):

• Starts at $(0, 0)$
• Peaks at $y = 1$, troughs at $y = -1$
• Repeats every $2\pi$

Cosine Graph ($y = \cos x$):

• Starts at $(0, 1)$
• Peaks at $y = 1$, troughs at $y = -1$
• Repeats every $2\pi$

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Worked Example

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Summary

• Sine and Cosine Graphs: Oscillate with amplitude 1 and period $2\pi$
• Amplitude: Maximum displacement from the midline.
• Period: The interval after which the function repeats.
• Applications: Useful in modelling waves, vibrations, and other periodic behaviours. Understanding the sine and cosine graphs provides a foundation for analysing complex waveforms and periodic phenomena.