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Worked solution

Normalize a square root before using the binomial series

If θ\theta is sufficiently small for θ3\theta^3 and higher powers of θ\theta to be neglected, find an approximation for 1+cos⁡θ\sqrt{1+\cos\theta}.

CJC · Lecture · Self-Practice 4

Factor the constant inside the square root

Both copies print the answer without working. Independently use cos⁡θ=1−θ2/2+O(θ4)\cos\theta=1-\theta^2/2+O(\theta^4):

1+cos⁡θ=2(1−θ24+O(θ4))1/2.\begin{aligned}&\sqrt{1+\cos\theta}\\ & =\sqrt2\left(1-\frac{\theta^2}{4}+O(\theta^4)\right)^{1/2}.\end{aligned}

The positive square root is intended; keep the factor √2 outside.

How did this part go?

Keep the quadratic correction

Apply the binomial coefficient 1/21/2 to the inner quadratic term:

1+cos⁡θ≈2[1+12(−θ24)]=2(1−θ28),\sqrt{1+\cos\theta}\approx\sqrt2\left[1+\frac12\left(-\frac{\theta^2}{4}\right)\right]=\sqrt2\left(1-\frac{\theta^2}{8}\right),

matching the full printed answer. Both radicals are preserved: one in the ask and the √2 prefactor in the answer.

How did this part go?

Common mistakes

Factoring 2 from inside a square root produces √2, and the inner −θ²/4 is multiplied by the binomial coefficient 1/2.

Solution notes

Both source radicals directly viewed; native extraction drops them but remains unaltered. Printed answer retained, working independently supplied. No marks or attribution.

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