{ ((sinx+siny=(3/2))),((2^(sin x) +2^(sin y) =2+(√2))) :} faind x=?


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Question Number 165995 by mathlove last updated on 11/Feb/22
${ ((sinx+siny=(3/2))),((2^(sin x) +2^(sin y) =2+(√2))) :} faind x=?$
${\begin{cases} s i n x + s i n y = \frac{3}{2} \\ 2^{\sin x} + 2^{\sin y} = 2 + \sqrt{2} \end{cases} f a i n d x = ?$
	Answered by alephzero last updated on 11/Feb/22
	${ ((sin x + cos y = (3/2))),((2^(sin x) + 2^(cos y) = 2 + (√2))) :} ⇒ { ((2^(sin x) = 2)),((2^(cos y) = (√2))) :} ∨ { ((2^(sin x) = (√2))),((2^(cos y) = 2)) :} ⇒ (x, y) = ((π/2) + kπ, { (((π/3) + 2kπ)),((((5π)/3) + 2kπ)) :}) ∨ ∨ (x, y) = ( { (((π/6) + 2kπ)),((((5π)/6) + 2kπ)) :}, 2kπ) sin (π/2) + cos (π/3) = 1 + (1/2) = (3/2) ✓ sin (π/6) + cos 2π = (1/2) + 1 = (3/2) ✓$
	${\begin{cases} \sin x + \cos y = \frac{3}{2} \\ 2^{\sin x} + 2^{\cos y} = 2 + \sqrt{2} \end{cases}$ $\Rightarrow {\begin{cases} 2^{\sin x} = 2 \\ 2^{\cos y} = \sqrt{2} \end{cases} \lor {\begin{cases} 2^{\sin x} = \sqrt{2} \\ 2^{\cos y} = 2 \end{cases}$ $\Rightarrow (x, y) = (\frac{π}{2} + k π, {\begin{cases} \frac{π}{3} + 2 k π \\ \frac{5 π}{3} + 2 k π \end{cases}) \lor$ $\lor (x, y) = ({\begin{cases} \frac{π}{6} + 2 k π \\ \frac{5 π}{6} + 2 k π \end{cases}, 2 k π)$ $\sin \frac{π}{2} + \cos \frac{π}{3} = 1 + \frac{1}{2} = \frac{3}{2} ✓$ $\sin \frac{π}{6} + \cos 2 π = \frac{1}{2} + 1 = \frac{3}{2} ✓$
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