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Find the  3 smallest positive x-intercepts of the graph of $$y = \cos(12x) + \cos(14x)$$ and list them in increasing order.

Apr 19, 2020

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Find the 3 smallest positive x-intercepts of the graph of $$y = \cos(12x) + \cos(14x)$$ and list them in increasing order.

Formula: $$\boxed{\cos(x) + \cos(y) = 2\cos\left(\dfrac{x+y}{2}\right)\cos\left(\dfrac{x-y}{2}\right) }$$

$$\begin{array}{|rcll|} \hline && \mathbf{\cos(12x) + \cos(14x)} \\ &=& \cos(14x) + \cos(12x) \\ &=& 2\cos\left(\dfrac{14x+12x}{2}\right)\cos\left(\dfrac{14x-12x}{2}\right) \\ &=& 2\cos\left(\dfrac{26x}{2}\right)\cos\left(\dfrac{2x}{2}\right) \\ &=& \mathbf{2\cos(13x)\cos(x)} \\ \hline \end{array}$$

x-intercepts of the graph

$$\begin{array}{|rcll|} \hline \mathbf{2\cos(13x)\cos(x)} &=& 0 \quad | \quad : 2 \\ \cos(13x)\cos(x) &=& 0 \\ \hline \mathbf{\cos(x)} &=& \mathbf{0} \\ x &=& \pm \arccos(0) +2k\pi \qquad k\in \mathbb{Z} \\ x &=& \pm \dfrac{\pi}{2} +2k\pi \\\\ x &=& \mathbf{+} \dfrac{\pi}{2}+2k\pi \qquad k = 0 \\ x &=& + \dfrac{\pi}{2} \\ \mathbf{x} &=& \mathbf{1.57079632679}\ \text{rad} \\\\ x &=& \mathbf{-} \dfrac{\pi}{2} +2k\pi \qquad k = 1 \\ x &=& - \dfrac{\pi}{2} +2 \pi \\ \mathbf{x} &=& \mathbf{4.71238898038}\ \text{rad} \\ \hline \mathbf{\cos(13x)} &=& \mathbf{0} \\ 13x &=& \pm \arccos(0) +2k\pi \qquad k\in \mathbb{Z} \\ 13x &=& \pm \dfrac{\pi}{2} +2k\pi \\\\ 13x &=& \mathbf{+}\dfrac{\pi}{2} +2k\pi \\ x &=& \dfrac{\pi}{2*13}+\dfrac{2\pi k}{13} \qquad k = 0 \\ x &=& \dfrac{\pi}{26} \\ \mathbf{x} &=& \mathbf{0.12083048668}\ \text{rad} \\\\ x &=& \mathbf{+}\dfrac{\pi}{26}+\dfrac{2\pi k}{13} \qquad k = 1 \\ x &=& \dfrac{\pi}{26}+\dfrac{2\pi}{13} \\ x &=& \dfrac{5\pi}{26} \\ \mathbf{x} &=& \mathbf{0.60415243338}\ \text{rad} \\\\ 13x &=& \mathbf{-}\dfrac{\pi}{2} +2k\pi \\ x &=& -\dfrac{\pi}{2*13}+\dfrac{2\pi k}{13} \qquad k = 1 \\ x &=& -\dfrac{\pi}{26}+\dfrac{2\pi}{13} \\ x &=& \dfrac{3\pi}{26} \\ \mathbf{x} &=& \mathbf{0.36249146003}\ \text{rad} \\ \hline \end{array}$$

The 3 smallest positive x-intercepts of the graph:
$$\mathbf{\dfrac{\pi}{26} },\ \mathbf{\dfrac{3\pi}{26} },\ \mathbf{\dfrac{5\pi}{26} }$$

Apr 20, 2020