Deduce the positive exact value of cos(π/12) from the factorisation.

01 / Standard#Your turn

Official paper · jm02-2026 · 4(b)(ii) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Expand the two quadratic factors as a difference of squares.

Hint 2

Compare the coefficient of z².

Worked solution

Let c=cos(π/12); the product has this middle coefficient.
$(z^2+1)^2-4c^2z^2=z^4+(2-4c^2)z^2+1$
Match coefficients and choose the positive square root because π/12 is acute.
$2-4c^2=-\sqrt3\implies c=\frac{\sqrt{2+\sqrt3}}2$

cos(π/12)=√(2+√3)/2.

Checks and common pitfalls: The outer square root covers 2+√3 together.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Let c=cos(π/12); the product has this middle coefficient.
$(z^2+1)^2-4c^2z^2=z^4+(2-4c^2)z^2+1$
Match coefficients and choose the positive square root because π/12 is acute.
$2-4c^2=-\sqrt3\implies c=\frac{\sqrt{2+\sqrt3}}2$

Think first. Reveal a hint when the class is ready.

2026 JM02