 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2017-resit/1/a/solution

(cos θ + i sin θ)^{3} = cos 3 θ + i sin 3 θ

. Expanding the cube and equating its real part gives

cos 3 θ = cos^{3} θ - 3 cos θ sin^{2} θ = 4 cos^{3} θ - 3 cos θ .

The last form uses

sin^{2} θ + cos^{2} θ = 1

; comparing the imaginary parts similarly gives the corresponding sine identity.

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