Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTRIGONOMETRIC FUNCTIONS1 Mark
Question
Find the general solution for the equation: cos 3x + cos x – cos 2x = 0
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Answer
Given: cos 3x + cos x - cox 2x = 0 $\Rightarrow$ $2 \cos \frac{(3 x+x)}{2} \cos \frac{3 x-x}{2}-\cos 2 x=0\left(\because \cos A+\cos B=2 \cos \frac{a+b}{2} \cos \frac{A-B}{2}\right)$ $\Rightarrow$ 2 cos 2 x cos x- cos 2 x = 0 $\Rightarrow$ cos 2 x(2 cos x-1) = 0 $\Rightarrow$ cos 2 x = 0 or 2 cos x-1 = 0 $\Rightarrow$ cos 2 x = 0 or cos x = 1/2 We know that the general solution for cos x = 0 is x = (2 n + 1) $\pi$/ 2, where n $\in$ Z and Z is set of integers. Therefore, 2 x = (2n + 1) $\frac{\pi}{2}$ or cos x = cos $\frac{\pi}{3}$, where n $\in$ Z and Z is set of integers $\Rightarrow$ x = (2n +1) $\frac{\pi}{4}$ or x = 2 n $\pi \pm \frac{\pi}{3}$, where n $\in$ Z and z is set of integers
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