Math Problem Statement
Solution
The graph provided is a plot of the cosine function over the interval . The task is to match the letters A through G to specific values of (in radians): .
Key points to analyze:
- starts at its maximum value of 1 when and completes one full cycle by .
- Critical points of the cosine function include:
- , (cosine maximum),
- , (crosses the x-axis),
- , (cosine minimum),
- , (crosses the x-axis again),
- , (returns to maximum).
Steps to match letters:
- A appears to correspond to , the maximum point where , but in the provided image, it could also correspond to some offset near 0.
- B seems to correspond to (first x-intercept, where ).
- C is around , where (the minimum point of the cosine curve).
- D might correspond to , the second x-intercept (where ).
- G likely corresponds to , as it is the maximum point again at the end of the cycle.
Now, to match values such as 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 and .
Tentative Matching:
- A: Could be either near or .
- B: Likely (1.57).
- C: Likely (3.14).
- D: Likely (4.71).
- E: Around or slightly less.
- F: Around .
- G: Likely .
Do you want more precise reasoning or any clarifications on this approach?
Here are 5 related questions:
- What is the general form of the cosine function?
- What are the critical points of a cosine function over one period?
- How do you calculate the cosine of non-standard angles like or ?
- How does the phase of a cosine wave affect its graph?
- 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
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