Exemplo 12 de Algebra and Trigonometry, 8.1 Graphs of the Sine and Cosine Functions

Finding the Vertical Component of Circular Motion

Problema e solução em inglês, como no livro original.

A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P, as shown in Figure 23. Sketch a graph of the height above the ground of the point $P$ as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.

An illustration of a circle lifted 4 feet off the ground. Circle has radius of 3 ft. There is a point P labeled on the circle's circumference.

Figure 23

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

Solução
Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in Figure 24.

Figure 24

Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let’s use a cosine function because it starts at the highest or lowest value, while a sine function starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.
Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.
Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that

$y = - 3 \cos (x) + 4$

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A weight is attached to a spring that is then hung from a board, as shown in Figure 25. As the spring oscillates up and down, the position $y$ of the weight relative to the board ranges from $–1$ in. (at time $x = 0)$ to $–7$ in. (at time $x = π)$ below the board. Assume the position of $y$ is given as a sinusoidal function of $x .$ Sketch a graph of the function, and then find a cosine function that gives the position $y$ in terms of $x .$

An illustration of a spring with length y.

Figure 25

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Figure 25

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Solução

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