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Assuming x is small, find an approximation for:

sin(x/2)cosx + sec(2x).

The answer I've been given is 1 + x/2 + 2x². I've tried substituing in approximations for sinx and cosx where required, and using compound angle formula, but keep on getting an x³ term . Any help would be greatly appreciated.

sin(x/2)cosx + sec(2x).

The answer I've been given is 1 + x/2 + 2x². I've tried substituing in approximations for sinx and cosx where required, and using compound angle formula, but keep on getting an x³ term . Any help would be greatly appreciated.

SsEe

If you get the correct answer but with x³ and higher terms you just say that those are small enough to be ignored.

sin(x/2)cosx + sec(2x).

≈ (x/2)(1 - x²/2) + (1 + 2x²)

expanding and ignoring x³ and higher:

sin(x/2)cosx + sec(2x) ≈ 1 + x/2 + 2x²

sin(x/2)cosx + sec(2x).

≈ (x/2)(1 - x²/2) + (1 + 2x²)

expanding and ignoring x³ and higher:

sin(x/2)cosx + sec(2x) ≈ 1 + x/2 + 2x²

Oh, of course ! Thanks for your help.

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