Problem 12 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the period and amplitude. $$y=2 \cos 3 x$$

Step-by-Step Solution

Answer

The amplitude is 2, and the period is \(2\pi / 3\).

Step 1: Identify Amplitude and B

From the given function \(y=2 \cos 3 x\), it can be identified that the amplitude, A is 2 and the value of B is 3. The amplitude is the maximum distance from the maximum of a wave to its equilibrium position

Step 2: Calculate Amplitude

The amplitude is equal to the absolute value of A. Thus, the amplitude here is \(|2|\) which equals 2. This is the maximum distance the wave reaches from its equilibrium position.

Step 3: Calculate Period

The period of the function is given by the formula \(T = 2\pi/B\). Substituting the value of B=3 in this formula gives \(T = 2\pi /3\).

Key Concepts

Amplitude

Amplitude is an essential concept when studying trigonometric functions like the cosine function. It represents the maximum extent of a wave measured from its equilibrium position. Essentially, it's the height of the wave.

• The amplitude is found by looking at the coefficient in front of the cosine function. In the equation \(y = 2 \cos 3x\), the amplitude is 2.
• To determine the amplitude, always take the absolute value of the coefficient. Therefore, even if the coefficient were negative, the amplitude remains positive.

Amplitude indicates how much the graph of the cosine function stretches vertically. If you visualize a wave, this is how high or low it goes from the center line (equilibrium). Understanding amplitude helps us predict and interpret real-world phenomena like sound waves or tides better.

Period of a Function

The period of a trigonometric function measures how long it takes the function to complete one full cycle before it repeats again. For functions like sine and cosine, this is particularly important.To find the period of a trigonometric function of the form \(y = A \cos(Bx)\) or \(y = A \sin(Bx)\):

• Use the formula \(T = \frac{2\pi}{B}\).
• The period is effectively dictated by the value of \(B\), the coefficient of \(x\) in the function.

In the given function \(y = 2 \cos 3x\), substituting \(B = 3\) into the formula gives a period \(T = \frac{2\pi}{3}\). This means the wave repeats every \(\frac{2\pi}{3}\) units along the x-axis.Understanding the period helps us know how frequently the pattern repeats, which is crucial in fields such as engineering and physics where periodic repetition is common.

Cosine Function

The cosine function is one of the primary trigonometric functions, often denoted as \(\cos(x)\). It is fundamental in mathematics and emerges often in geometry, physics, and engineering.The cosine function has several key traits:

• It is periodic, meaning it repeats its values in regular intervals or periods.
• The general form of the cosine function is \(y = A \cos(Bx + C) + D\) where:
• \(A\) is the amplitude, affecting the wave's height.
• \(B\) determines the period as explained earlier.
• \(C\) shifts the graph left or right (horizontal shift).
• \(D\) moves the graph up or down (vertical shift).

In \(y = 2 \cos 3x\), it has an amplitude of 2 and a period of \(\frac{2\pi}{3}\), with no horizontal or vertical shift.Cosine functions are used to model oscillatory movements and can describe phenomena such as light waves, sound waves, and seasonal temperature variations.