Showing that x cos ⁡ φ + y sin ⁡ φ < 2

woowheedr

woowheedr

Answered

2022-07-01

Showing that x cos φ + y sin φ < 2

Answer & Explanation

treccinair

treccinair

Expert

2022-07-02Added 18 answers

Each of the quantities x , y , cos φ , sin φ is at most 1 in magnitude, so x cos φ + y sin φ 1 1 + 1 1 = 2
The only way to achieve equality is if x and y both have magnitude 1, which cannot happen, so the inequality is actually strict: x cos φ + y sin φ < 2
In fact, as Martin Argerami points out, one can use Cauchy-Schwarz to obtain the tight bound x cos φ + y sin φ 1

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