Question

Determine the amplitude and period of each function without graphing. $$ y=3 \cos x $$

Short answer

Amplitude: 3, Period: \ 2\pi

Step-by-step solution

Identify the Standard Form

The standard form of a cosine function is given by \[ y = a \cos(bx) \]. In this problem, the function given is \[ y = 3 \cos(x) \].

Determine the Amplitude

The amplitude of a cosine function is the absolute value of the coefficient \(a\) in front of the cosine term. Here, \( a = 3 \). Therefore, the amplitude is \[ |a| = |3| = 3 \].

Determine the Period

The period of a cosine function is determined by the value of \(b\) in \( y = a \cos(bx) \). The period is given by \[ \text{Period} = \frac{2\pi}{|b|} \]. In this case, \( b = 1 \). Therefore, the period is \[ \text{Period} = \frac{2\pi}{|1|} = 2\pi \].

Key concepts

Amplitude of Cosine Function

Understanding the amplitude of the cosine function is essential in mastering trigonometric functions. The amplitude refers to the height of the wave from the centerline to its peak. For the cosine function in its standard form, \( y = a \cos(bx) \), the amplitude is given by the absolute value of the coefficient \( a \) in front of the cosine term. This coefficient determines how tall or short the waves are.

For instance, in the function \( y = 3 \cos(x) \), the coefficient \( a \) is 3. Taking the absolute value, the amplitude is \( |3| = 3 \). This means the graph of this function will oscillate 3 units above and 3 units below the centerline (which is the x-axis).

Knowing the amplitude helps in graphing the cosine function and understanding its behavior, especially when comparing multiple trigonometric functions.

Period of Cosine Function

The period of a cosine function defines how long it takes for the wave to complete one full cycle before it repeats itself. For the standard form \( y = a \cos(bx) \), the period can be calculated using the formula:

• \( \text{Period} = \frac{2\pi}{|b|} \)

Here, \( b \) is the coefficient inside the cosine function that affects the rate at which the function cycles.

Taking the given function \( y = 3 \cos(x) \), we see that \( b \) is 1. Plugging into the formula:

• \( \text{Period} = \frac{2\pi}{|1|} = 2\pi \)

This reveals that the function completes one full cycle every \( 2\pi \) units along the x-axis. The understanding of the period is crucial for graphing since it tells us how frequently the function repeats its values.

Trigonometric Functions

Trigonometric functions, including the cosine function, are fundamental in mathematics. They describe relationships between angles and sides of triangles and are vital in periodic functions. The primary trigonometric functions are sine, cosine, and tangent.

The cosine function \( \cos(x) \) is especially important because of its wave-like properties, which repeat at regular intervals. The amplitude and period determine the shape and frequency of the wave respectively.

• Amplitude: How high or low the wave goes.


• Period: How long before the wave pattern starts repeating.

Learning about these properties helps in various applications, from engineering to physics, where wave behaviors need to be understood and predicted. Mastering trigonometric functions provides a strong foundation for more advanced mathematical concepts.