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Solve the Following Equation: Cos X + Cos 2 X + Cos 3 X = 0 - Mathematics

Sum

Solve the following equation:

\[\cos x + \cos 2x + \cos 3x = 0\]
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Solution

\[\cos x + \cos 2x + \cos 3x = 0\]

Now,

\[(\cos x + \cos3x) + \cos2x = 0\]
\[ \Rightarrow 2 \cos \left( \frac{4x}{2} \right) \cos \left( \frac{2x}{2} \right) + \cos2x = 0\]
\[ \Rightarrow 2 \cos2x \cos x + \cos2x = 0\]
\[ \Rightarrow \cos2x ( 2 \cos x + 1) = 0\]

\[\Rightarrow \cos 2x = 0\] or,
\[2 \cos x + 1 = 0\]
\[\Rightarrow \cos 2x = \cos \frac{\pi}{2}\] or
\[\cos x = - \frac{1}{2} = \cos \frac{2\pi}{3}\]
\[\Rightarrow 2x = (2n + 1) \frac{\pi}{2}\],
\[n \in Z\] or

\[x = 2m\pi \pm \frac{2\pi}{3}, m \in Z\]

\[\Rightarrow x = (2n + 1)\frac{\pi}{4}, n \in Z\]
\[x = 2m\pi \pm \frac{2\pi}{3}, m \in Z\]
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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 11 Trigonometric equations
Exercise 11.1 | Q 4.1 | Page 22
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