Explanation
Definition
The cosine function describes the horizontal projection of a point on the unit circle. It is periodic and central in trigonometry.
f(x) = \cos(x)
Procedure
Scheme
Function term: Write the term and determine values, e.g., cosine of 45.
f(x) = \cos(x)
Value progression: Examine the behavior of the cosine function. Observe that \cos(0) = 1 and \cos(\pi) = -1 holds.
Period Length = 2\pi
Determine f(0) and f(\pi) at f(x) = \cos(x): f(0) = 1,\; f(\pi) = -1.
Zeros: Solve \cos(x) = 0 with x = \tfrac{\pi}{2} + k\pi and determine the zeros in the interval [0,\,2\pi].
\cos(x) = 0 \Rightarrow x = \tfrac{\pi}{2} + k\pi
Find the zeros of f(x) = \cos(x) in the interval [0,\,2\pi]: The zeros are: x = \tfrac{\pi}{2} and x = \tfrac{3\pi}{2}.
Misunderstandings
Common Errors
- ★It is often forgotten to convert angles to radians.
- ★It is often mistakenly assumed that cosine 45 = 0.45 instead of \frac{\sqrt{2}}{2}.
- ★It is often not recognized correctly that the function is periodic.
Problems
1 / 3
Calculate
f(0) at f(x) = \cos(x)
2 / 3
Draw the graph of
f(x) = \cos(x) in the interval [0,\,2\pi]
3 / 3
Find all zeros of
f(x) = \cos(x) in the interval [0,\,4\pi]
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