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Which one of the following is the correct graphical representation for functions, sin(x) and sin2(x) for \(0 \le x \le \pi\)?

  • A.Graph of sin(x) as a solid curve and sin squared of x as a dotted curve from 0 to pi, both peaking at 1 at pi/2, with the dotted curve inside the solid curve
  • B.Graph of sin(x) as a solid curve and a dotted curve labelled sin squared of x that starts above zero and lies above the solid curve
  • C.Graph of sin(x) as a solid curve peaking at 1 and a dotted sin squared of x curve with a lower peak, below the solid curve
  • D.Graph of sin(x) as a solid curve peaking at 1 and a dotted curve labelled sin squared of x that starts high, dips to 1 at pi/2 and rises again

Answer: A

The correct pair of graphs is the one in option (A).

On \(0 \le x \le \pi\) the sine is between 0 and 1, and squaring a number between 0 and 1 makes it smaller. So \(\sin^{2}x \le \sin x\) everywhere, the two curves meet only at \(x = 0\), \(\pi/2\) and \(\pi\), and both reach exactly 1 at \(\pi/2\). Near 0 and \(\pi\) the square is much flatter, because \(\sin^{2}x \approx x^{2}\) there.

Option (B) shows \(\sin^{2}x\) starting above zero at \(x = 0\), but \(\sin^{2}0 = 0\). Option (C) has the square peaking below 1, but \(\sin^{2}(\pi/2) = 1\). Option (D) puts \(\sin^{2}x\) above \(\sin x\) and above 1, which can never happen.

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