 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2002/paper-1/11/b/solution

Euler's formula and its version with

- θ

give

cos θ = \frac{e ^{i θ} + e ^{- i θ}}{2}, sin θ = \frac{e ^{i θ} - e ^{- i θ}}{2 i} .

Squaring

e^{i θ} = cos θ + i sin θ

and comparing real and imaginary parts with

e^{2 i θ}

cos 2 θ = cos^{2} θ - sin^{2} θ = 2 cos^{2} θ - 1 = 1 - 2 sin^{2} θ, sin 2 θ = 2 sin θ cos θ .

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