Question

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7sinx+17cosx=√338

Answer 1

$7sinx+17cosx=\sqrt{338}\; \Big |\; :\sqrt{338}\\\\\star \; \; 7^2+17^2=338\; \; \; \; \to \; \; \; \; \Big(\dfrac{7}{\sqrt{338}}\Big)^2+\Big(\dfrac{17}{\sqrt{338}}\Big)^2=1\; \; \; \; \to \\\\cos\varphi=\dfrac{7}{\sqrt{338}}\; ,\; \; sin\varphi=\dfrac{17}{\sqrt{338}}\; \; ,\; \; \varphi=arctg\dfrac{17}{7}\; \; \star \\\\\\\dfrac{7}{\sqrt{338}}\cdot sinx +\dfrac{17}{\sqrt{338}}\cdot cosx=1\\\\\\cos\varphi\cdot sinx+sin\varphi\cdot cosx=1\\\\sin(x+\varphi)=1\\\\x+\varphi=\dfrac{\pi}{2}+2\pi n\; ,\; n\in Z$

$x=\varphi+\dfrac{\pi}{2}+2\pi n\; ,\; n\in Z\\\\\\x=arctg\dfrac{17}{7}+\dfrac{\pi}{2}+2\pi n\; ,\; n\in Z$