Discussion

The value of \(\sin^2 1° + \sin^2 2° + ... + \sin^2 89°\) is:

Answer: C

We pair the terms using the identity \(\sin(90-A) = \cos A\).

The series is \((\sin^2 1° + \sin^2 89°) + (\sin^2 2° + \sin^2 88°) + ... + \sin^2 45°\).

Each pair \((\sin^2 A + \sin^2(90-A))\) becomes \((\sin^2 A + \cos^2 A) = 1\).

There are 44 such pairs (from 1° to 44°). So the sum of these pairs is 44.

The middle term left over is \(\sin^2 45° = (1/\sqrt{2})^2 = 1/2 = 0.5\).

Total sum = 44 + 0.5 = 44.5.

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