Bounds Math Example 4

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Example 4

hard
Prove that \(\sin(x) \leq 1\) and \(\sin(x) \geq -1\) for all \(x\) using the unit circle definition.

Solution

  1. 1
    On the unit circle, \(\sin(x)\) equals the y-coordinate of a point.
  2. 2
    The unit circle has radius 1, so all y-coordinates satisfy \(-1 \leq y \leq 1\).
  3. 3
    Therefore \(-1 \leq \sin(x) \leq 1\) for all real \(x\).
  4. 4
    The bounds \(\pm 1\) are achieved at \(x = \pi/2\) and \(x = -\pi/2\).

Answer

\(-1 \leq \sin(x) \leq 1\) for all \(x\)
The unit circle constrains y-coordinates to \([-1,1]\). Since \(\sin(x)\) is a y-coordinate on the unit circle, it is bounded by ±1.

About Bounds

The upper and lower limits within which a quantity must lie; often expressed as a≤x≤ba \leq x \leq b.

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More Bounds Examples

Example 1 medium

For (f(x) = -x^2 + 6x - 5), find the maximum value (upper bound) on ([0, 5]).

Example 2 hard

Show that for all real (x), (x^2 geq 0). What is the greatest lower bound (infimum) of (x^2)?

Example 3 medium

For (g(x) = 3x + 1) on ([0, 4]), find the minimum and maximum values.

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