Trigonometric equations

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Number of values of $θ$ for the equation $\cos θ + \cos 3 θ + \cos 5 θ + \cos 7 θ = 0$ in $(0, π)$ is

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$W e h a v e, \cos θ + \cos 3 θ + \cos 5 θ + \cos 7 θ = 0$
$\Rightarrow (\cos θ + \cos 7 θ) + (\cos 3 θ + \cos 5 θ) = 0$
$\Rightarrow 2 \cos 4 θ \cos 3 θ + 2 \cos 4 θ . \cos θ = 0$
$\Rightarrow \cos 4 θ (\cos 3 θ + \cos θ) = 0$
$\Rightarrow \cos 4 θ (2 \cos 2 θ \cos θ) = 0$
Either $\cos θ = 0$ $\Rightarrow θ = (2 n + 1) \frac{π}{2}$
$o r \cos 2 θ = 0 \Rightarrow θ = (2 n + 1) \frac{π}{4}$
$o r \cos 4 θ = 0 \Rightarrow θ = (2 n + 1) \frac{π}{8}$
$S i n c e θ \in (0, π) \Rightarrow θ = \frac{π}{8}, \frac{π}{4}, \frac{3 π}{8}, \frac{π}{2}, \frac{5 π}{8}, \frac{3 π}{4}, \frac{7 π}{8}$

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Trigonometric equations

Remember concepts with our Masterclasses.

Number of values of θ for the equation cosθ+cos3θ+cos5θ+cos7θ=0 in 0,π is

We have, cosθ+cos3θ+cos5θ+cos7θ=0⇒cosθ+cos7θ+cos3θ+cos5θ=0⇒2cos4θcos3θ+2cos4θ.cosθ=0⇒cos4θcos3θ+cosθ=0⇒cos4θ2cos2θcosθ=0Either cosθ=0⇒θ=2n+1π2 or cos2θ=0⇒θ=2n+1π4or cos4θ=0⇒θ=2n+1π8Since θ∈0,π ⇒θ=π8,π4,3π8,π2,5π8,3π4,7π8

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Number of values of $θ$ for the equation $\cos θ + \cos 3 θ + \cos 5 θ + \cos 7 θ = 0$ in $(0, π)$ is