Find the general solution of the equation 3^(sin 2x + 2 cos^2 x) + 3^(1 − sin 2x + 2 sin^2 x) = 28


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Question Number 14632 by Tinkutara last updated on 03/Jun/17

$Find the general solution of the$ $equation$ $3^{\sin 2 x + 2 \cos^{2} x} + 3^{1 - \sin 2 x + 2 \sin^{2} x} = 28$
	Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 03/Jun/17

	$s i n 2 x + 2 {c o s}^{2} x = m \Rightarrow 1 - s i n 2 x + 2 {s i n}^{2} x =$ $= 1 - s i n 2 x + 2 - 2 {c o s}^{2} x = 3 - m$ $\Rightarrow 3^{m} + 3^{3 - m} = 28 \Rightarrow (3^{m} = t)$ $t^{2} - 28 t + 27 = 0 \Rightarrow t = 1, 27$ $1) 3^{m} = 1 \Rightarrow s i n 2 x + 2 {c o s}^{2} x - 1 = 0 \Rightarrow$ $s i n 2 x + c o s 2 x = 0 \Rightarrow t g 2 x = 0 \Rightarrow x = \frac{k π}{2}$ $(c o s 2 x \neq 0 \Rightarrow 2 x \neq 2 l π + \frac{π}{2} \Rightarrow x \neq l π + \frac{π}{4})$ $2) 3^{m} = 27 = 3^{3} \Rightarrow m = 3$ $\Rightarrow s i n 2 x + 2 {c o s}^{2} x = 3 \Rightarrow s i n 2 x + c o s 2 x = 2$ $\Rightarrow \sqrt{2} s i n (2 x + \frac{π}{4}) = 2 \Rightarrow s i n (2 x + \frac{π}{4}) = \sqrt{2}$ $i t i s i m p o s s i b l e \overset{}{.}$
	Commented by Tinkutara last updated on 04/Jun/17

	$Thanks Sir!$
	Answered by 433 last updated on 03/Jun/17

	$3^{s i n 2 x + 2 (1 - {s i n}^{2} x)} + 3^{1 - s i n 2 x + 2 {s i n}^{2} x} = 28$ $3^{s i n 2 x - 2 {s i n}^{2} x + 2} + 3^{1 - s i n 2 x + 2 {s i n}^{2} x} = 28$ $3^{s i n 2 x - 2 {s i n}^{2} x} = y > 0$ $y \times 3^{2} + \frac{3}{y} = 28$ $9 y^{2} - 28 y + 3 = 0$ $Δ = 28^{2} - 4 \times 9 \times 3 = 676 = 26^{2}$ $y_{1, 2} = \frac{28 \pm 26}{18} = \frac{1}{9} o r 3$ $3^{s i n 2 x - 2 {s i n}^{2} x} = \frac{1}{9}$ $s i n 2 x - 2 {s i n}^{2} x = - 2$ $2 s i n x c o s x - 2 (1 - {c o s}^{2} x) = - 2$ $2 s i n x c o s x + 2 {c o s}^{2} x = 0$ $2 c o s x (s i n x + c o s x) = 0$ $c o s x = 0 \Leftrightarrow x = 2 k π \pm \frac{π}{2}$ $s i n x = - c o s x \Leftrightarrow - (s i n - x) = - c o s x$ $s i n (- x) = s i n (\frac{π}{2} - x)$ $- x = 2 k π + \frac{π}{2} - x \Leftrightarrow 2 k π + π / 2 = 0$ $- x = 2 k π + x + \frac{π}{2} \Leftrightarrow x = k π - \frac{π}{4}$ $3^{s i n 2 x - 2 {s i n}^{2} x} = 3$ $s i n 2 x - 2 {s i n}^{2} x = 1$ $2 s i n x c o s x - 2 (1 - {c o s}^{2} x) = 1$ $2 c o s x (s i n x + c o s x) = 3$ $s i n (2 x + \frac{π}{4}) = \frac{\sqrt{2}}{2} (s i n 2 x + c o s 2 x) =$ $\frac{\sqrt{2}}{2} (2 s i n x c o s x + {c o s}^{2} x - {s i n}^{2} x)$ $s i n (2 x + \frac{π}{4}) \times \sqrt{2} + 1 = 2 s i n x c o s x + {c o s}^{2} x - {s i n}^{2} x + 1 =$ $2 s i n x c o s x + 2 {c o s}^{2} x = 2 c o s x (s i n x + c o s x)$ $\Rightarrow s i n (2 x + \frac{π}{4}) \times \sqrt{2} + 1 = 3$ $s i n (2 x + \frac{π}{4}) = \sqrt{2}$
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