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Solve $$\cos^7(x) + \cos^7\left(x+\dfrac{2\pi}{3}\right) + \cos^7\left(x+\dfrac{4\pi}{3}\right)=0$$ for $$0\leq x\leq 2\pi$$

Jun 11, 2020

#1
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Solve

$$\cos^7(x) + \cos^7\left(x+\dfrac{2\pi}{3}\right) + \cos^7\left(x+\dfrac{4\pi}{3}\right)=0$$ for $$0\leq x\leq 2\pi$$

Formula:

$$\begin{array}{|rcll|} \hline \cos^{n}(x) &=& {\frac {1}{2^{n}}}\,\sum \limits_{k=0}^{n}{n \choose k}\,\cos\Big((n-2k)x\Big);\quad n\in \mathbb {N} \\\\ \cos^{7}(x) &=& {\frac {1}{2^{7}}}\,\sum \limits_{k=0}^{7}{7 \choose k}\,\cos\Big((7-2k)x\Big) \\ \hline \mathbf{\cos ^{7}(x)} &=& \mathbf{{ \frac {1}{2^{7}} } \Big( \cos(7x)+7\cos(5x) +21\cos(3x)+35\cos(x) \Big)} \\\\ \cos^{7}\left(x+\dfrac{2\pi}{3}\right) &=& { \frac {1}{2^{7}} } \Bigg( \cos(7\left(x+\dfrac{2\pi}{3}\right))+7\cos(5\left(x+\dfrac{2\pi}{3}\right)) +21\cos(3\left(x+\dfrac{2\pi}{3}\right))+35\cos\left(x+\dfrac{2\pi}{3}\right) \Bigg) \\ &=& { \frac {1}{2^{7}} } \Bigg( \cos\left(7x+\dfrac{14\pi}{3}\right) +7\cos\left(5x+\dfrac{10\pi}{3}\right) +21\cos(3x+2\pi)+35\cos\left(x+\dfrac{2\pi}{3}\right) \Bigg) \\ \mathbf{\cos^{7}\left(x+\dfrac{2\pi}{3}\right)} &=& \mathbf{{ \frac {1}{2^{7}} } \Bigg( \cos\left(7x+\dfrac{14\pi}{3}\right) +7\cos\left(5x+\dfrac{10\pi}{3}\right) +21\cos(3x)+35\cos\left(x+\dfrac{2\pi}{3}\right) \Bigg) } \\\\ \cos^{7}\left(x+\dfrac{4\pi}{3}\right) &=& { \frac {1}{2^{7}} } \Big( \cos(7\left(x+\dfrac{4\pi}{3}\right))+7\cos(5\left(x+\dfrac{4\pi}{3}\right)) +21\cos(3\left(x+\dfrac{4\pi}{3}\right))+35\cos\left(x+\dfrac{4\pi}{3}\right) \Bigg) \\ &=& { \frac {1}{2^{7}} } \Bigg( \cos\left(7x+\dfrac{28\pi}{3}\right) +7\cos\left(5x+\dfrac{20\pi}{3}\right) +21\cos(3x+4\pi)+35\cos\left(x+\dfrac{4\pi}{3}\right) \Bigg) \\ \mathbf{\cos^{7}\left(x+\dfrac{4\pi}{3}\right)} &=& \mathbf{{ \frac {1}{2^{7}} } \Bigg( \cos\left(7x+\dfrac{28\pi}{3}\right) +7\cos\left(5x+\dfrac{20\pi}{3}\right) +21\cos(3x)+35\cos\left(x+\dfrac{4\pi}{3}\right) \Bigg) } \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline && \cos^7(x) + \cos^7\left(x+\dfrac{2\pi}{3}\right) + \cos^7\left(x+\dfrac{4\pi}{3}\right) \\ &=& \frac {1}{128} \Bigg( 61\cos(3x) \\ && + \cos(7x) + \cos\left(7x+\dfrac{14\pi}{3}\right) + \cos\left(7x+\dfrac{28\pi}{3}\right)\\ && + 7\Big( \cos(5x) + \cos\left(5x+\dfrac{10\pi}{3}\right) + \cos\left(5x+\dfrac{20\pi}{3}\right) \Big) \\ && +35\Big( \cos(x) + \cos\left(x+\dfrac{2\pi}{3}\right) + \cos\left(x+\dfrac{4\pi}{3}\right) \Big) \Bigg) \\ \hline \end{array}$$

$$\text{Formula}: \cos x+\cos y=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}$$

$$\begin{array}{|rcll|} \hline && \mathbf{\cos(7x) + \cos\left(7x+\dfrac{14\pi}{3}\right) + \cos\left(7x+\dfrac{28\pi}{3}\right)} \\\\ &=& \cos(7x) + 2\cos\left(\dfrac{14x+14\pi}{2}\right) \cos\left(\dfrac{\dfrac{14\pi}{3}}{2}\right) \\\\ &=& \cos(7x) + 2\cos(7x+7\pi) \cos\left(\dfrac{7\pi}{3}\right) \\\\ &=& \cos(7x) + 2\cos(7x+\pi) \cos\left(\dfrac{7\pi}{3}\right) \\\\ &=& \cos(7x) + 2\Big( \cos(7x)\cos(\pi)-\sin(7x)\sin(\pi) \Big) \cos\left(\dfrac{7\pi}{3}\right) \\\\ &=& \cos(7x) - 2\cos(7x) \cos\left(\dfrac{7\pi}{3}\right) \\\\ &=& \cos(7x) \Big(1 - 2\cos\left(\dfrac{7\pi}{3}\right)\Big) \quad | \quad \cos\left(\dfrac{7\pi}{3}\right)=\dfrac{1}{2} \\\\ &=& \cos(7x) (1 - 2*\dfrac{1}{2}) \\\\ &=& \cos(7x) (1 - 1) \\\\ &=& \mathbf{ 0 } \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline && \mathbf{7\Bigg( \cos(5x) + \cos\left(5x+\dfrac{10\pi}{3}\right) + \cos\left(5x+\dfrac{20\pi}{3}\right) \Bigg)} \\\\ &=& 7\Bigg(\cos(5x) + 2\cos\left(\dfrac{10x+10\pi}{2}\right) \cos\left(\dfrac{\dfrac{10\pi}{3}}{2}\right) \Bigg)\\\\ &=& 7\Bigg(\cos(5x) + 2\cos(5x+5\pi) \cos\left(\dfrac{5\pi}{3}\right) \Bigg)\\\\ &=& 7\Bigg(\cos(5x) + 2\cos(5x+\pi) \cos\left(\dfrac{5\pi}{3}\right) \Bigg)\\\\ &=& 7\Bigg(\cos(5x) + 2\Big( \cos(5x)\cos(\pi)-\sin(5x)\sin(\pi) \Big) \cos\left(\dfrac{5\pi}{3}\right)\Bigg) \\\\ &=& 7\Bigg(\cos(5x) - 2\cos(5x) \cos\left(\dfrac{5\pi}{3}\right) \Bigg)\\\\ &=& 7\Bigg(\cos(5x) \Big(1 - 2\cos\left(\dfrac{5\pi}{3}\right)\Big) \quad | \quad \cos\left(\dfrac{5\pi}{3}\right)=\dfrac{1}{2} \Bigg)\\\\ &=& 7\Bigg(\cos(5x) (1 - 2*\dfrac{1}{2})\Bigg) \\\\ &=& 7\Bigg(\cos(5x) (1 - 1)\Bigg) \\\\ &=& \mathbf{ 0 } \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline && \mathbf{35\Bigg( \cos(x) + \cos\left(x+\dfrac{2\pi}{3}\right) + \cos\left(x+\dfrac{4\pi}{3}\right) \Bigg)} \\\\ &=& 35\Bigg(\cos(x) + 2\cos\left(\dfrac{2x+2\pi}{2}\right) \cos\left(\dfrac{\dfrac{2\pi}{3}}{2}\right) \Bigg)\\\\ &=& 35\Bigg(\cos(x) + 2\cos(x+\pi) \cos\left(\dfrac{1\pi}{3}\right) \Bigg)\\\\ &=& 35\Bigg(\cos(x) + 2\Big( \cos(x)\cos(\pi)-\sin(x)\sin(\pi) \Big) \cos\left(\dfrac{1\pi}{3}\right)\Bigg) \\\\ &=& 35\Bigg(\cos(x) - 2\cos(x) \cos\left(\dfrac{1\pi}{3}\right) \Bigg)\\\\ &=& 35\Bigg(\cos(x) \Big(1 - 2\cos\left(\dfrac{1\pi}{3}\right)\Big) \quad | \quad \cos\left(\dfrac{1\pi}{3}\right)=\dfrac{1}{2} \Bigg)\\\\ &=& 35\Bigg(\cos(x) (1 - 2*\dfrac{1}{2})\Bigg) \\\\ &=& 35\Bigg(\cos(x) (1 - 1)\Bigg) \\\\ &=& \mathbf{ 0 } \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline && \mathbf{\cos^7(x) + \cos^7\left(x+\dfrac{2\pi}{3}\right) + \cos^7\left(x+\dfrac{4\pi}{3}\right)} \\ &=& \frac {1}{128} \Bigg( 61\cos(3x) + 0+0+0 \Bigg) \\ &=& \mathbf{\frac {61}{128}*\cos(3x)} \\ \hline \end{array}$$

$$\begin{array}{|rclcrcl|} \hline \mathbf{\cos^7(x) + \cos^7\left(x+\dfrac{2\pi}{3}\right) + \cos^7\left(x+\dfrac{4\pi}{3}\right)} &=& \mathbf{0} \\\\ \frac {61}{128}*\cos(3x) &=& 0 \\\\ \cos(3x) &=& 0 \\ 3x &=& \pm\arccos(0) +2\pi n \quad n \in \mathbb{Z} \\\\ 3x &=& \pm\dfrac{\pi}{2} +2\pi n \\\\ \mathbf{x} &=& \mathbf{\pm\dfrac{\pi}{6} +\dfrac{2\pi n}{3} } \\ \hline \begin{array}{|rclcrcl|} \hline x&=& \dfrac{\pi}{6} +\dfrac{2\pi n}{3} &\text{or}& x&=& -\dfrac{\pi}{6} +\dfrac{2\pi n}{3} \\\\ \mathbf{x_1}&=& \mathbf{\dfrac{\pi}{6}} & & x_4&=& -\dfrac{\pi}{6} +\dfrac{2\pi}{3} \\\\ & & & & \mathbf{x_4}&=& \mathbf{\dfrac{\pi}{2}} \\\\ x_2&=& \dfrac{\pi}{6} +\dfrac{2\pi}{3} & & x_5&=& -\dfrac{\pi}{6} +\dfrac{4\pi}{3} \\\\ \mathbf{x_2}&=& \mathbf{\dfrac{5\pi}{6}} & & \mathbf{x_5}&=& \mathbf{\dfrac{7\pi}{6}} \\\\ x_3&=& \dfrac{\pi}{6} +\dfrac{4\pi}{3} & & x_6&=& -\dfrac{\pi}{6} +\dfrac{6\pi}{3} \\\\ \mathbf{x_3}&=& \mathbf{\dfrac{3\pi}{2}} & & \mathbf{x_6}&=& \mathbf{\dfrac{11\pi}{6}} \\ \hline \end{array} \\ \hline \end{array}$$

Solutions $$0\leq x\leq 2\pi$$:

$$\begin{array}{rcll} x &= \dfrac{\pi}{6} \\\\ x &= \dfrac{\pi}{2} \\\\ x &= \dfrac{5\pi}{6} \\\\ x &= \dfrac{7\pi}{6} \\\\ x &= \dfrac{3\pi}{2} \\\\ x &= \dfrac{11\pi}{6} \\ \end{array}$$

Jun 11, 2020
edited by heureka  Jun 11, 2020
edited by heureka  Jun 11, 2020
edited by heureka  Jun 11, 2020