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Answer 1

$\sin^{3} x + \cos^{3} x = {(\sin x + \cos x)}^{3} - 3 \sin x . \cos x . (\sin x + \cos x) Đ ặ t t = \sin x + \cos x = \sqrt{2} . \sin (x + \frac{π}{4}) (t \in [- \sqrt{2}; \sqrt{2}]) = > t^{2} = 1 + \sin 2 x = > \sin 2 x = t^{2} - 1 \sin x . \cos x = - \frac{1}{2} + \frac{t^{2}}{2}$

$= > C ó : t^{3} - 3 . (\frac{t^{2}}{2} - \frac{1}{2}) . t = \frac{\sqrt{3}}{2} . t^{2} < = > t^{3} - \frac{3}{2} t^{3} - \frac{\sqrt{3}}{2} t^{2} + \frac{3}{2} t = 0 < = > \frac{- 1}{2} t^{3} - \frac{\sqrt{3}}{2} t^{2} + \frac{3}{2} t = 0 < = > t = 0 h o ặ c t = \frac{\sqrt{15} - \sqrt{3}}{2} (d o t \in [- \sqrt{2}; \sqrt{2}]) t = 0 = > \sqrt{2} . \sin (x + \frac{π}{4}) = 0 < = > \sin (x + \frac{π}{4}) = 0 < = > x + \frac{π}{4} = k π < = > x = \frac{- π}{4} + kπ (k \in Z) t = \frac{\sqrt{15} - \sqrt{3}}{2} < = > \sqrt{2} . \sin (x + \frac{π}{4}) = \frac{\sqrt{15} - \sqrt{3}}{2} < = > \sin (x + \frac{π}{4}) = \frac{\sqrt{30} - \sqrt{6}}{4} < = > x = a r c \sin (\frac{\sqrt{30} - \sqrt{6}}{4}) + k 2 π hặc x = π - \arcsin (\frac{\sqrt{30} - \sqrt{6}}{4}) + k 2 π, k \in Z$ .

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Answer 2

$\sin^{3} x + \cos^{3} x = {(\sin x + \cos x)}^{3} - 3 \sin x . \cos x . (\sin x + \cos x) Đ ặ t t = \sin x + \cos x = \sqrt{2} . \sin (x + \frac{π}{4}) (t \in [- \sqrt{2}; \sqrt{2}]) = > t^{2} = 1 + \sin 2 x = > \sin 2 x = t^{2} - 1 \sin x . \cos x = - \frac{1}{2} + \frac{t^{2}}{2}$

$= > C ó : t^{3} - 3 . (\frac{t^{2}}{2} - \frac{1}{2}) . t = \frac{\sqrt{3}}{2} . t^{2} < = > t^{3} - \frac{3}{2} t^{3} - \frac{\sqrt{3}}{2} t^{2} + \frac{3}{2} t = 0 < = > \frac{- 1}{2} t^{3} - \frac{\sqrt{3}}{2} t^{2} + \frac{3}{2} t = 0 < = > t = 0 h o ặ c t = \frac{\sqrt{15} - \sqrt{3}}{2} (d o t \in [- \sqrt{2}; \sqrt{2}]) t = 0 = > \sqrt{2} . \sin (x + \frac{π}{4}) = 0 < = > \sin (x + \frac{π}{4}) = 0 < = > x + \frac{π}{4} = k π < = > x = \frac{- π}{4} + kπ (k \in Z) t = \frac{\sqrt{15} - \sqrt{3}}{2} < = > \sqrt{2} . \sin (x + \frac{π}{4}) = \frac{\sqrt{15} - \sqrt{3}}{2} < = > \sin (x + \frac{π}{4}) = \frac{\sqrt{30} - \sqrt{6}}{4} < = > x = a r c \sin (\frac{\sqrt{30} - \sqrt{6}}{4}) + k 2 π hặc x = π - \arcsin (\frac{\sqrt{30} - \sqrt{6}}{4}) + k 2 π, k \in Z$ .

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