6. Rezolvaṭi următoarele ecuatii trigonometrice elementare: \( \begin{array}{ll}\text { a) } \sin 5 x=\sin 3 x ; & \text { b) } \cos x=\cos 4 x \\ \text { c) } \cos 5 x=-\sin 3 x ; & \text { d) } \operatorname{tg} 4 x=\operatorname{ctg}\left(3 x+\frac{\pi}{4}\right) ; \\ \begin{array}{ll}\text { e) } \sin \left(\frac{3 \pi}{2}+x\right)+\cos 3 x=0 ; & \text { f) } \cos \left(\frac{3 x}{2}+\frac{\pi}{4}\right)+\sin \left(\frac{2 x}{3}-\frac{\pi}{4}\right)=0\end{array} \\ \text { 7. Rezolvați următoarele ecuații trigonometrice elementare: } \\ \begin{array}{lll}\text { a) } \sin x=\sin 7 x ; & \text { b) }-\sin 2 x=\sin x ; & \text { e) } \operatorname{tg} 4 x=\operatorname{tg} 2 x ;\end{array} & \text { c) } \cos 4 x=\cos 5 x \\ \text { d) } \cos 2 x=-\cos 6 x ; & \text { f) } \operatorname{ctg} 2 x=\operatorname{ctg} x\end{array} \)

Solve the equation by following steps: - step0: Solve for \(x\): \(2x-x=0\) - step1: Subtract the terms: \(x=0\) Solve the equation \( \text{tg}(4x) - \text{tg}(2x) = 0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(4x-2x=0\) - step1: Subtract the terms: \(2x=0\) - step2: Rewrite the expression: \(x=0\) Solve the equation \( \cos(4x) - \cos(5x) = 0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\cos\left(4x\right)-\cos\left(5x\right)=0\) - step1: Transform the expression: \(-2\sin\left(\frac{9x}{2}\right)\sin\left(-\frac{x}{2}\right)=0\) - step2: Elimination the left coefficient: \(\sin\left(\frac{9x}{2}\right)\sin\left(-\frac{x}{2}\right)=0\) - step3: Separate into possible cases: \(\begin{align}&\sin\left(\frac{9x}{2}\right)=0\\&\sin\left(-\frac{x}{2}\right)=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=\frac{2k\pi }{9},k \in \mathbb{Z}\\&x=2k\pi ,k \in \mathbb{Z}\end{align}\) - step5: Find the union: \(x=\frac{2k\pi }{9},k \in \mathbb{Z}\) Solve the equation \( \sin(5x) - \sin(3x) = 0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sin\left(5x\right)-\sin\left(3x\right)=0\) - step1: Transform the expression: \(2\cos\left(4x\right)\sin\left(x\right)=0\) - step2: Elimination the left coefficient: \(\cos\left(4x\right)\sin\left(x\right)=0\) - step3: Separate into possible cases: \(\begin{align}&\cos\left(4x\right)=0\\&\sin\left(x\right)=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=\frac{\pi }{8}+\frac{k\pi }{4},k \in \mathbb{Z}\\&x=k\pi ,k \in \mathbb{Z}\end{align}\) - step5: Find the union: \(x=\left\{ \begin{array}{l}k\pi \\\frac{\pi }{8}+\frac{k\pi }{4}\end{array}\right.,k \in \mathbb{Z}\) Solve the equation \( \cos(5x) + \sin(3x) = 0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\cos\left(5x\right)+\sin\left(3x\right)=0\) - step1: Transform the expression: \(2\sin\left(\frac{16x+\pi }{4}\right)\cos\left(\frac{4x+\pi }{4}\right)=0\) - step2: Elimination the left coefficient: \(\sin\left(\frac{16x+\pi }{4}\right)\cos\left(\frac{4x+\pi }{4}\right)=0\) - step3: Separate into possible cases: \(\begin{align}&\sin\left(\frac{16x+\pi }{4}\right)=0\\&\cos\left(\frac{4x+\pi }{4}\right)=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=\frac{3\pi }{16}+\frac{k\pi }{4},k \in \mathbb{Z}\\&x=\frac{\pi }{4}+k\pi ,k \in \mathbb{Z}\end{align}\) - step5: Find the union: \(x=\left\{ \begin{array}{l}\frac{3\pi }{16}+\frac{k\pi }{4}\\\frac{\pi }{4}+k\pi \end{array}\right.,k \in \mathbb{Z}\) Solve the equation \( \text{tg}(4x) - \text{ctg}(3x + \frac{\pi}{4}) = 0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(4x-\left(3x+\frac{\pi }{4}\right)=0\) - step1: Subtract the terms: \(x-\frac{\pi }{4}=0\) - step2: Move the constant to the right side: \(x=0+\frac{\pi }{4}\) - step3: Add the terms: \(x=\frac{\pi }{4}\) Solve the equation \( \cos(x) - \cos(4x) = 0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\cos\left(x\right)-\cos\left(4x\right)=0\) - step1: Transform the expression: \(-2\sin\left(\frac{5x}{2}\right)\sin\left(-\frac{3x}{2}\right)=0\) - step2: Elimination the left coefficient: \(\sin\left(\frac{5x}{2}\right)\sin\left(-\frac{3x}{2}\right)=0\) - step3: Separate into possible cases: \(\begin{align}&\sin\left(\frac{5x}{2}\right)=0\\&\sin\left(-\frac{3x}{2}\right)=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=\frac{2k\pi }{5},k \in \mathbb{Z}\\&x=\frac{2k\pi }{3},k \in \mathbb{Z}\end{align}\) - step5: Find the union: \(x=\left\{ \begin{array}{l}\frac{2k\pi }{5}\\\frac{2k\pi }{3}\end{array}\right.,k \in \mathbb{Z}\) Solve the equation \( \sin(x) - \sin(7x) = 0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(\sin\left(x\right)-\sin\left(7x\right)=0\) - step1: Transform the expression: \(2\cos\left(4x\right)\sin\left(-3x\right)=0\) - step2: Elimination the left coefficient: \(\cos\left(4x\right)\sin\left(-3x\right)=0\) - step3: Separate into possible cases: \(\begin{align}&\cos\left(4x\right)=0\\&\sin\left(-3x\right)=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=\frac{\pi }{8}+\frac{k\pi }{4},k \in \mathbb{Z}\\&x=\frac{k\pi }{3},k \in \mathbb{Z}\end{align}\) - step5: Find the union: \(x=\left\{ \begin{array}{l}\frac{k\pi }{3}\\\frac{\pi }{8}+\frac{k\pi }{4}\end{array}\right.,k \in \mathbb{Z}\) Solve the equation \( -\cos(2x) - \cos(6x) = 0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(-\cos\left(2x\right)-\cos\left(6x\right)=0\) - step1: Transform the expression: \(-2\cos\left(x+3x\right)\cos\left(-2x\right)=0\) - step2: Elimination the left coefficient: \(\cos\left(x+3x\right)\cos\left(-2x\right)=0\) - step3: Separate into possible cases: \(\begin{align}&\cos\left(x+3x\right)=0\\&\cos\left(-2x\right)=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=\frac{\pi }{8}+\frac{k\pi }{4},k \in \mathbb{Z}\\&x=\frac{\pi }{4}+\frac{k\pi }{2},k \in \mathbb{Z}\end{align}\) - step5: Find the union: \(x=\left\{ \begin{array}{l}\frac{\pi }{8}+\frac{k\pi }{4}\\\frac{\pi }{4}+\frac{k\pi }{2}\end{array}\right.,k \in \mathbb{Z}\) Solve the equation \( -\sin(2x) - \sin(x) = 0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(-\sin\left(2x\right)-\sin\left(x\right)=0\) - step1: Transform the expression: \(-2\sin\left(\frac{2x+x}{2}\right)\cos\left(\frac{x}{2}\right)=0\) - step2: Elimination the left coefficient: \(\sin\left(\frac{2x+x}{2}\right)\cos\left(\frac{x}{2}\right)=0\) - step3: Separate into possible cases: \(\begin{align}&\sin\left(\frac{2x+x}{2}\right)=0\\&\cos\left(\frac{x}{2}\right)=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=\frac{2k\pi }{3},k \in \mathbb{Z}\\&x=\pi +2k\pi ,k \in \mathbb{Z}\end{align}\) - step5: Find the union: \(x=\left\{ \begin{array}{l}\frac{2k\pi }{3}\\\pi +2k\pi \end{array}\right.,k \in \mathbb{Z}\) Iată soluțiile pentru ecuațiile trigonometrice date: ### 6. Rezolvarea ecuațiilor trigonometrice: a) \( \sin 5x = \sin 3x \) Soluția este: \[ x = \left\{ \begin{array}{l} k\pi \\ \frac{\pi}{8} + \frac{k\pi}{4} \end{array} \right., k \in \mathbb{Z} \] b) \( \cos x = \cos 4x \) Soluția este: \[ x = \left\{ \begin{array}{l} \frac{2k\pi}{5} \\ \frac{2k\pi}{3} \end{array} \right., k \in \mathbb{Z} \] c) \( \cos 5x = -\sin 3x \) Soluția este: \[ x = \left\{ \begin{array}{l} \frac{3\pi}{16} + \frac{k\pi}{4} \\ \frac{\pi}{4} + k\pi \end{array} \right., k \in \mathbb{Z} \] d) \( \tan 4x = \cot\left(3x + \frac{\pi}{4}\right) \) Soluția este: \[ x = \frac{\pi}{4} \] e) \( \sin\left(\frac{3\pi}{2} + x\right) + \cos 3x = 0 \) Aici a apărut o eroare în procesarea ecuației. Voi încerca să o rezolv din nou. f) \( \cos\left(\frac{3x}{2} + \frac{\pi}{4}\right) + \sin\left(\frac{2x}{3} - \frac{\pi}{4}\right) = 0 \) Aici a apărut o eroare în procesarea ecuației. Voi încerca să o rezolv din nou. ### 7. Rezolvarea ecuațiilor trigonometrice: a) \( \sin x = \sin 7x \) Soluția este: \[ x = \left\{ \begin{array}{l} \frac{k\pi}{3} \\ \frac{\pi}{8} + \frac{k\pi}{4} \end{array} \right., k \in \mathbb{Z} \] b) \( -\sin 2x = \sin x \) Soluția este: \[ x = \left\{ \begin{array}{l} \frac{2k\pi}{3} \\ \pi + 2k\pi \end{array} \right., k \in \mathbb{Z} \] c) \( \tan 4x = \tan 2x \) Soluția este: \[ x = 0 \] d) \( \cos 4x = \cos 5x \) Soluția este: \[ x = \frac{2k\pi}{9}, k \in \mathbb{Z} \] e) \( -\cos 2x = -\cos 6x \) Soluția este: \[ x = 0 \] f) \( \cot 2x = \cot x \) Soluția este: \[ x = \left\{ \begin{array}{l} \frac{\pi}{8} + \frac{k\pi}{4} \\ \frac{\pi}{4} + \frac{k\pi}{2} \end{array} \right., k \in \mathbb{Z} \] Dacă ai nevoie de detalii suplimentare sau de clarificări pentru anumite ecuații, te rog să îmi spui!