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The cosine function, often represented as \( \cos(x) \), is a fundamental aspect of trigonometry. It is defined for all real numbers and represents the x-coordinate of a point on the unit circle.
In a typical cosine wave, the function starts at its maximum value, decreases to its minimum, and then returns to the maximum, forming a smooth, repetitive wave.
For the equation \( y = -\cos 2x \), we notice that the cosine function is affected by a negative sign, indicating that the wave is inverted. This causes the wave to start at its minimum value, rise to zero, spike to its maximum, and repeat. The presence of the term \( 2x \) implies that the wave is horizontally compressed, altering its frequency and period.

Wave frequency refers to how often the wave patterns repeat in a given interval. It is a measure of how "tight" or "loose" the waves are on the x-axis.
In the context of trigonometric functions like cosines, frequency is determined by the coefficient of \( x \) inside the function.
For our function \( y = -\cos 2x \), the coefficient "2" signifies that the wave repeats its cycle twice as fast as the standard cosine function.
Therefore, the frequency is said to be "2," indicating that the wave completes two full cycles over a length of \( 2\pi \).

• Higher frequencies result in more cycles over the same distance.
• Lower frequencies create broader, more stretched-out waves.

Amplitude is the height of the wave from its central axis to its peak or trough. It measures the wave's strength or intensity.
In the general cosine function \( y = a \cos(bx) \), the amplitude is given by the absolute value of \( a \).
For the equation \( y = -\cos 2x \), the amplitude can be identified as 1, although being negative indicates wave inversion. It does not affect the actual height but reflects the wave orientation.
This means the wave oscillates between -1 and 1, centered around the x-axis.

• An amplitude of 1 signifies moderate fluctuation from the center line.
• Larger amplitudes encompass taller waves.

Periodicity in trigonometric functions refers to the interval at which the wave pattern fully repeats itself. It defines the cycle length of the wave.
The formula to calculate the period of a cosine function is \( T = \frac{2\pi}{b} \), where \( b \) is the frequency component.
In \( y = -\cos 2x \), substituting \( b \) with 2 gives a period \( T = \frac{2\pi}{2} = \pi \).
This indicates that the pattern of the wave - from minimum to maximum and back again - completes every \( \pi \) units along the x-axis.

• Shorter periods result in quicker oscillation cycles.
• Longer periods mean the cycles are more spread out.