nice-calculus-if-a-b-c-0-and-acos-2-x-bsin-2-x-c-then-prove-that-a-cos-2-x-b-sin-2-x-c-

Question Number 133443 by mnjuly1970 last updated on 22/Feb/21

\dots \dots n i c e c a l c u l u s \dots \dots .

i f a, b, c ⩾ 0

a n d :: {a c o s}^{2} (x) + {b s i n}^{2} (x) ⩽ c

t h e n p r o v e t h a t ::

\sqrt{a} {c o s}^{2} (x) + \sqrt{b} {s i n}^{2} (x) ⩽ \sqrt{c}

\dots \dots \dots \dots .

Answered by mnjuly1970 last updated on 22/Feb/21

{\vec{u}}_{1} = (\sqrt{a} c o s (x), \sqrt{b} s i n (x)) \in R^{2}

{\vec{u}}_{2} = (c o s (x), s i n (x)) \in R^{2}

∣ {\vec{u}}_{1} . {\vec{u}}_{2} ∣ \underset{s c h w a r t z}{\overset{c a u c h y}{⩽}} ∣ {\vec{u}}_{1} ∣∣ {\vec{u}}_{2} ∣

∣ \sqrt{a} {c o s}^{2} (x) + \sqrt{b} {s i n}^{2} (x) ∣⩽ \sqrt{{a c o s}^{2} (x) + {b s i n}^{2} (x)} . \sqrt{[{c o s}^{2} (x) + {s i n}^{2} (x)] = 1}

∣ \sqrt{a} {c o s}^{2} (x) + \sqrt{b} {s i n}^{2} (x) ∣⩽ \sqrt{c}

\dots . .