woowheedr

2022-07-01

Showing that $x\mathrm{cos}\phi +y\mathrm{sin}\phi <2$

treccinair

Expert

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

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