First slide

Trigonometric equations

Question

General solution of the equation $1 + \cos 3 θ = 2 \cos 2 θ$ is

Moderate

A

$θ = n π \pm \frac{π}{6}, x = 2 n π$
B

$θ = 2 n π \pm \frac{π}{6}, x = n π$
C

$θ = n π \pm \frac{π}{3}, x = \frac{n π}{2}$
D

$θ = n π \pm \frac{π}{4}, x = n π$

Solution

$\begin{array}{l} 1 + \cos 3 θ = 2 \cos 2 θ \\ \Rightarrow 1 + 4 \cos^{3} θ - 3 \cos θ = 2 (2 \cos^{2} θ - 1) \\ \Rightarrow 4 \cos^{3} θ - 4 \cos^{2} θ - 3 \cos θ + 3 = 0 \\ \Rightarrow (\cos θ - 1) (4 \cos^{2} θ - 3) = 0 \\ \Rightarrow \cos θ = 1 o r \cos^{2} θ = \frac{3}{4} = {(\frac{\sqrt{3}}{2})}^{2} = \cos^{2} (\frac{π}{6}) \\ \Rightarrow θ = 2 n π o r θ = n π \pm \frac{π}{6}, n \in z \end{array}$

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