[tex]\boxed{\left [ \begin{array}{ccc} x = \pi n, n \in \mathbb Z \\ \\ x = \dfrac{\pi n}{2} , n \in \mathbb Z \\ \\ x = \dfrac{\pi n}{6}, n \in \mathbb Z \end{array}\right}[/tex]
[tex]\left [ \begin{array}{ccc} x = \pi n, n \in \mathbb Z \\ 2x = \pi n, n \in \mathbb Z|:2 \\ 6x = \pi n, n \in \mathbb Z|:6 \end{array}\right[/tex]
[tex]\left [ \begin{array}{ccc} x = \pi n, n \in \mathbb Z \\ \\ x = \dfrac{\pi n}{2} , n \in \mathbb Z \\ \\ x = \dfrac{\pi n}{6}, n \in \mathbb Z \end{array}\right[/tex]
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Ответ:
[tex]\boxed{\left [ \begin{array}{ccc} x = \pi n, n \in \mathbb Z \\ \\ x = \dfrac{\pi n}{2} , n \in \mathbb Z \\ \\ x = \dfrac{\pi n}{6}, n \in \mathbb Z \end{array}\right}[/tex]
Формулы:
[tex]\cos \alpha - \cos \beta = -2 \sin \bigg (\dfrac{\alpha + \beta }{2} \bigg) \sin \bigg (\dfrac{\alpha - \beta }{2} \bigg)[/tex]
[tex]\sin \alpha + \sin \beta = 2 \sin \bigg (\dfrac{\alpha + \beta }{2} \bigg) \cos \bigg (\dfrac{\alpha - \beta }{2} \bigg)[/tex]
[tex]\sin x = 0 \Longrightarrow x = \pi n, n \in \mathbb Z[/tex]
Объяснение:
[tex]\cos 3x - \cos 5x + \cos 7x - \cos 9x = 0|\cdot(-1)[/tex]
[tex]\cos 5x - \cos 3x + \cos 9x - \cos 7x = 0[/tex]
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а)
[tex]\cos 5x - \cos 3x = -2 \sin \bigg (\dfrac{5x + 3x }{2} \bigg) \sin \bigg (\dfrac{5x - 3x }{2} \bigg) = -2 \sin \bigg (\dfrac{8x }{2} \bigg) \sin \bigg (\dfrac{2x }{2} \bigg) =[/tex]
[tex]= -2 \sin 4x \sin x[/tex]
б)
[tex]\cos 9x - \cos 7x = -2 \sin \bigg (\dfrac{9x + 7x }{2} \bigg) \sin \bigg (\dfrac{9x - 7x}{2} \bigg) =-2 \sin \bigg (\dfrac{16x }{2} \bigg) \sin \bigg (\dfrac{2x }{2} \bigg) =[/tex]
[tex]= -2 \sin 8x \sin x[/tex]
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[tex]-2 \sin 4x \sin x -2 \sin 8x \sin x = 0| \cdot (-0,5)[/tex]
[tex]\sin 4x \sin x + \sin 8x \sin x = 0[/tex]
[tex]\sin x(\sin 8x + \sin 4x) = 0[/tex]
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в)
[tex]\sin 8x + \sin 4x = 2 \sin \bigg (\dfrac{8x + 4x}{2} \bigg) \cos \bigg (\dfrac{8x - 4x}{2} \bigg) = 2 \sin \bigg (\dfrac{12x}{2} \bigg) \cos \bigg (\dfrac{4x}{2} \bigg) =[/tex]
[tex]= 2 \sin 6x \sin 2x[/tex]
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[tex]2 \sin 6x \sin 2x \sin x = 0|:2[/tex]
[tex]\sin x \sin 2x \sin 6x = 0 \Longleftrightarrow \left [ \begin{array}{ccc} \sin x = 0 \\ \sin 2x = 0 \\ \sin 6x = 0 \end{array}\right[/tex]
[tex]\left [ \begin{array}{ccc} x = \pi n, n \in \mathbb Z \\ 2x = \pi n, n \in \mathbb Z|:2 \\ 6x = \pi n, n \in \mathbb Z|:6 \end{array}\right[/tex]
[tex]\left [ \begin{array}{ccc} x = \pi n, n \in \mathbb Z \\ \\ x = \dfrac{\pi n}{2} , n \in \mathbb Z \\ \\ x = \dfrac{\pi n}{6}, n \in \mathbb Z \end{array}\right[/tex]