 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2018/1/b/solution

sin (x + i y) = sin x cosh y + i cos x sinh y .

∣ sin (x + i y) ∣^{2} = sin^{2} x cosh^{2} y + cos^{2} x sinh^{2} y = sin^{2} x + sinh^{2} y \geq sin^{2} x .

Taking nonnegative square roots proves

∣ sin z ∣ \geq ∣ sin x ∣

. Equality holds exactly when

y = 0

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