A trough is the lowest point or minimum value in a periodic wave or function, such as a sine or cosine curve. It represents the point where the wave or function reaches its lowest amplitude before beginning to rise again.
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The trough of a sine or cosine function represents the point where the function reaches its minimum value before beginning to increase again.
The distance between two consecutive troughs or peaks of a sine or cosine function is equal to the period of the function.
The amplitude of a sine or cosine function is measured from the midline to the peak or trough of the function.
The frequency of a sine or cosine function is inversely proportional to the period, with higher frequencies corresponding to shorter periods.
The shape and placement of the troughs in a sine or cosine graph can provide information about the phase and initial conditions of the function.
Review Questions
Explain the relationship between the trough of a sine or cosine function and the period of the function.
The distance between two consecutive troughs (or peaks) of a sine or cosine function is equal to the period of the function. The period represents the time it takes for one complete cycle of the function to occur, and the trough marks the lowest point of that cycle before the function begins to rise again. The placement of the troughs is directly related to the period and frequency of the sine or cosine function.
Describe how the amplitude of a sine or cosine function is related to the height of the troughs and peaks.
The amplitude of a sine or cosine function is measured from the midline to the peak or trough of the function. The distance between the midline and the trough (or peak) represents the amplitude of the function. A larger amplitude means the troughs and peaks are farther from the midline, while a smaller amplitude indicates the troughs and peaks are closer to the midline. The amplitude is a key characteristic that determines the overall size and scale of the sine or cosine function.
Analyze how the shape and placement of the troughs in a sine or cosine graph can provide information about the phase and initial conditions of the function.
The shape and placement of the troughs in a sine or cosine graph can reveal important information about the phase and initial conditions of the function. The phase of the function, which represents its position within the periodic cycle, is indicated by the horizontal positioning of the troughs. Additionally, the initial conditions, such as the starting amplitude and the point at which the function begins, are reflected in the vertical positioning and shape of the troughs. By carefully examining the troughs, one can gain valuable insights into the underlying characteristics and behavior of the sine or cosine function.