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Question

The number of solutions of the equation cosθ+cos4θ+cos7θ=0 in [0,2π] is

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Solution

cosθ+cos4θ+cos7θ=02cos4θcos3θ+cos4θ=0cos4θ=0 or 2cos3θ+1=0

For cos4θ=0
4θ=(2n+1)π2, nZθ=(2n+1)π8θ=π8, 3π8, 5π8, 7π8, 9π8, 11π8, 13π8, 15π8
8 solutions

For 2cos3θ+1=0
cos3θ=123θ=2nπ + 2π3 or 2nπ + 4π3, nZθ=2π9, 4π9, 8π9, 10π9, 14π9, 16π9
6 solutions

So, there are total (8+6)=14 solutions.

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