Math Problem Statement

Match each of the letters A-G in the cosine function graph to one of the following x-values (in radians): 1, 2, 4, 5, π/2, π, and 3π/2.

Solution

The graph provided is a plot of the cosine function y=cos(x)y = \cos(x) over the interval [0,2π][0, 2\pi]. The task is to match the letters A through G to specific values of xx (in radians): 1,2,4,5,π2,π,and3π21, 2, 4, 5, \frac{\pi}{2}, \pi, \text{and} \frac{3\pi}{2}.

Key points to analyze:

  • y=cos(x)y = \cos(x) starts at its maximum value of 1 when x=0x = 0 and completes one full cycle by x=2πx = 2\pi.
  • Critical points of the cosine function include:
    • x=0x = 0, y=1y = 1 (cosine maximum),
    • x=π2x = \frac{\pi}{2}, y=0y = 0 (crosses the x-axis),
    • x=πx = \pi, y=1y = -1 (cosine minimum),
    • x=3π2x = \frac{3\pi}{2}, y=0y = 0 (crosses the x-axis again),
    • x=2πx = 2\pi, y=1y = 1 (returns to maximum).

Steps to match letters:

  1. A appears to correspond to x=0x = 0, the maximum point where y=1y = 1, but in the provided image, it could also correspond to some offset near 0.
  2. B seems to correspond to x=π2x = \frac{\pi}{2} (first x-intercept, where y=0y = 0).
  3. C is around x=πx = \pi, where y=1y = -1 (the minimum point of the cosine curve).
  4. D might correspond to x=3π2x = \frac{3\pi}{2}, the second x-intercept (where y=0y = 0).
  5. G likely corresponds to x=2πx = 2\pi, as it is the maximum point again at the end of the cycle.

Now, to match values such as 1,2,4,and51, 2, 4, \text{and} 5 which are not standard cosine landmarks:

  • These values likely fall between the critical points and need to be estimated based on their positions on the curve relative to π2,π,3π2,\frac{\pi}{2}, \pi, \frac{3\pi}{2}, and 2π2\pi.

Tentative Matching:

  • A: Could be either near 11 or 2π2\pi.
  • B: Likely π2\frac{\pi}{2} (1.57).
  • C: Likely π\pi (3.14).
  • D: Likely 3π2\frac{3\pi}{2} (4.71).
  • E: Around 44 or slightly less.
  • F: Around 55.
  • G: Likely 2π2\pi.

Do you want more precise reasoning or any clarifications on this approach?

Here are 5 related questions:

  1. What is the general form of the cosine function?
  2. What are the critical points of a cosine function over one period?
  3. How do you calculate the cosine of non-standard angles like x=1x = 1 or x=4x = 4?
  4. How does the phase of a cosine wave affect its graph?
  5. Can you graph a cosine function with amplitude and phase shifts?

Tip: The cosine function is symmetric, so identifying points between its peaks and troughs can help predict values at other positions.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Cosine Function
Radians

Formulas

y = cos(x)

Theorems

Periodicity of the Cosine Function

Suitable Grade Level

Grades 9-12