The energy, w, of an inductor at time t is given by w=1-cos(50 t) Sketch the graph of w against

step 1 Hi friends here it is given through the inductor of inductance 50 micro henry that is 50 into 10

The energy, w, of an inductor at time t is given by...

Key Concepts

Cosine Function Properties

The cosine function is a fundamental trigonometric function that oscillates between -1 and 1 in a regular, periodic pattern. Understanding its behavior, including its maximum, minimum, and symmetry about the vertical axis, is essential for interpreting and graphing any function that incorporates cosine as part of its formulation.

Graph Transformations

Graph transformations involve modifying the basic shape or position of a function’s graph through operations such as vertical translation, horizontal scaling, and reflection. When a cosine function is adjusted by adding or subtracting constants, or when its argument is multiplied by a factor, these changes alter the amplitude, period, and vertical position of the graph, enabling a clearer depiction of the function’s behavior over a specified interval.

Angular Frequency and Period

The angular frequency in a trigonometric function, indicated by the multiplier of the independent variable, determines how quickly the function completes its cycles. The period of a function is inversely related to the angular frequency, which is critical in determining how the function repeats over time. Recognizing this relationship allows one to accurately predict the span of one complete cycle and ensures that key features of the oscillation are captured when sketching the graph.

Identifying Key Points in a Graph

Determining and plotting key points, such as intercepts, maximum and minimum values, and endpoints, is a core aspect of sketching a function’s graph. In the context of a transformed cosine function, this process involves calculating the function values at significant time intervals to highlight the pattern of oscillation, which aids in constructing an accurate representation of the energy variation or any similarly modeled phenomenon.

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