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Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides in the context of the unit circle. Cosine, sine, and tangent are the most familiar trigonometric functions. They are periodic, which means they repeat values in regular intervals. For example, the cosine function, denoted as \( \cos \), repeats its pattern every \(2\pi\) radians.

When represented on a graph, trigonometric functions produce waves with specific characteristics: amplitude, period, phase shift, and vertical shift. Understanding these traits is crucial when analyzing and comparing different trigonometric functions. Let's delve into the amplitude and period, which are the focus when comparing \(y=\frac{1}{2} \cos x\) and \(y=3 \cos 2x\).

The period of a cosine function refers to the length of the interval in which the function's values repeat. For the standard cosine function \(y = \cos x\), the period is \(2\pi\) radians. In other words, the function completes one full cycle and starts repeating its values after every \(2\pi\) units along the x-axis.

Impact of Coefficients on Period

When you have a cosine function in the form \(y = \cos(bx)\), the period is affected by the coefficient \(b\). To find the new period, you divide \(2\pi\) by the absolute value of \(b\). If \(b = 1\), as in our first function \(y=\frac{1}{2} \cos x\), the period remains \(2\pi\). However, if \(b = 2\), like in the second function \(y=3 \cos 2x\), the period is halved to \(\pi\), resulting in a function that repeats twice as often as the standard cosine function.

The amplitude of a cosine function signifies the maximum distance it reaches from its central axis (usually the x-axis) during each cycle. The standard amplitude is 1 for the base function \(y = \cos x\), reflecting the maximum height of the cosine wave above or below this axis.

Effect of Multiplication on Amplitude

When the function is multiplied by a coefficient, like \(y=a \cos x\), the amplitude is the absolute value of \(a\). For \(y=\frac{1}{2} \cos x\), the coefficient is \(\frac{1}{2}\), giving us an amplitude of \(\frac{1}{2}\). This means the wave peaks at \(\frac{1}{2}\) above and dips to \(\frac{1}{2}\) below the central axis. Comparatively, the function \(y=3 \cos 2x\) with a coefficient of 3 has an amplitude of 3, indicating much higher peaks and deeper troughs, three times the standard amplitude of the base cosine wave.