Tập nghiệm của phương trình sin3x + cos3x = 1 là:

5/10

S = \(\left\{ {k2\pi |k \in \mathbb{Z}} \right\}\);

S = \(\left\{ {k\pi |k \in \mathbb{Z}} \right\}\);

S = \(\left\{ {k\frac{2}{3}\pi ;x = \frac{\pi }{6} + k\frac{2}{3}\pi |k \in \mathbb{Z}} \right\}\);

S = \(\left\{ {k2\pi ;x = \frac{\pi }{6} + k2\pi |k \in \mathbb{Z}} \right\}\).

Giải thích

Đáp án đúng là: C

sin3x + cos3x = 1

⇔ \(\frac{1}{{\sqrt {1 + 1} }}\sin 3x + \frac{1}{{\sqrt {1 + 1} }}\cos 3x = \frac{1}{{\sqrt {1 + 1} }}\)

⇔ \(\frac{1}{{\sqrt 2 }}\sin 3x + \frac{1}{{\sqrt 2 }}\cos 3x = \frac{1}{{\sqrt 2 }}\)

⇔ \(\cos \frac{\pi }{4}.\sin 3x + \sin \frac{\pi }{4}.\cos 3x = \frac{1}{{\sqrt 2 }}\)

⇔ \(\sin \left( {3x + \frac{\pi }{4}} \right) = \frac{1}{{\sqrt 2 }}\)

⇔ \(\sin \left( {3x + \frac{\pi }{4}} \right) = \sin \frac{\pi }{4}\)

⇔ \(\left[ \begin{array}{l}3x + \frac{\pi }{4} = \frac{\pi }{4} + k2\pi \\3x + \frac{\pi }{4} = \pi - \frac{\pi }{4} + k2\pi \end{array} \right.\) , k ∈ ℤ

⇔ \(\left[ \begin{array}{l}3x = k2\pi \\3x = \frac{\pi }{2} + k2\pi \end{array} \right.\) , k ∈ ℤ

⇔ \(\left[ \begin{array}{l}x = k\frac{2}{3}\pi \\x = \frac{\pi }{6} + k\frac{2}{3}\pi \end{array} \right.\) , k ∈ ℤ.