Question

Multiple Choice One period of the graph of \(y=\sin (\omega x)\) or \(y=\cos (\omega x)\) is called \(\mathrm{a}(\mathrm{n})\) _______. (a) amplitude (b) phase shift (c) transformation (d) cycle

Short answer

Cycle

Step-by-step solution

Identify the graph

Focus on the graph functions provided: \(y = \sin(\omega x)\) and \(y = \cos(\omega x)\). These are sinusoidal functions (sine and cosine).

Define the period

The period of a sinusoidal function is the length of one complete cycle of the wave. For the standard trigonometric functions \(sin\) and \(cos\), a full period corresponds to one complete oscillation or wave.

Match the definition with the choices

Review the given options:(a) amplitude(b) phase shift(c) transformation(d) cycleThe correct term to describe one period of the graph is 'cycle'.

Key concepts

sine functions

Sine functions are one of the basic building blocks in trigonometry. The function is written as \( y = \sin(\theta) \), where \( \theta \) represents the angle. It produces a wave-like graph that oscillates between -1 and 1.
Some key properties of the sine function include:

• Amplitude: The maximum distance from the middle of the wave to its peak or trough (for standard sine, this is 1).
• Period: The length of one complete cycle of the wave (for standard sine, this is \( 2\pi \)).

cosine functions

Cosine functions are very similar to sine functions but start at their maximum value when \( x = 0 \). The function is written as \( y = \cos(\theta)\), where \( \theta \) represents the angle. The graph of the cosine function is also wave-like, oscillating between -1 and 1.
Essential features of cosine functions include:

• Amplitude: Like sine, this is the maximum distance from the middle of the wave to its peak or trough (for standard cosine, this is also 1).
• Period: The length of one complete cycle of the wave (for standard cosine, this is \( 2\pi \)).

period of a function

The period of a function is a key concept in trigonometry. It refers to the length of one complete cycle of a repeating function. For the standard sine and cosine functions, the period is \( 2\pi \). This means that every \( \theta = 2\pi \), the function starts repeating itself.

The concept of period is crucial when working with:

• Graphs: It helps to determine how often the wave pattern repeats.
• Equations: Understanding the period allows us to predict future values.

cycle in trigonometry

In trigonometry, a cycle refers to one complete oscillation of a periodic function, such as the sine or cosine function. It covers the distance between two equivalent points on the graph.
A few important points about cycles:

• Each cycle looks identical to the one before and after it.
• The concept of cycle helps in understanding the repeating nature of waves.
• In the context of sine and cosine functions, a complete cycle includes both a positive and negative peak.

When the problem mentions that one period of the graph is called a cycle, it highlights that after one period (\(2\pi \) for sine and cosine), the function's graph pattern starts again.