 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014-resit/11/v/solution

Conjugating the first vector gives

⟨ a, b ⟩ = i e^{- i x} + (1 - i) e^{in x}, Re ⟨ a, b ⟩ = sin x + cos (n x) + sin (n x) .

Their squared norms are

3

and

2

, independently of

x, n

cos θ = \frac{sin x + cos ( n x ) + sin ( n x )}{6} .

The complex Cauchy-Schwarz inequality ensures that this real quotient lies in

[- 1, 1]

 New to topics? Read the docs here!