resoudre-dans-pi-pi-cosx-cos2x-cos3x-0-

Question Number 198483 by Guillaume last updated on 20/Oct/23

r e s o u d r e d a n s] - π; π]

c o s x + c o s 2 x + c o s 3 x = 0

Answered by Frix last updated on 21/Oct/23

c = \cos x

c + 2 c^{2} - 1 + 4 c^{3} - 3 c = 0

c^{3} + \frac{c^{2}}{2} - \frac{c}{2} - \frac{1}{4} = 0

(c + \frac{\sqrt{2}}{2}) (c + \frac{1}{2}) (c - \frac{\sqrt{2}}{2}) = 0

x = \pm \frac{3 π}{4} \lor x = \pm \frac{2 π}{3} \lor x = \pm \frac{π}{4}

Answered by mr W last updated on 21/Oct/23

\cos x + 2 \cos^{2} x - 1 + 4 \cos^{3} x - 3 \cos x = 0

4 \cos^{3} x + 2 \cos^{2} x - 2 \cos x - 1 = 0

l e t u = \cos x

4 u^{3} + 2 u^{2} - 2 u - 1 = 0

(2 u + 1) (2 u^{2} - 1) = 0

\Rightarrow u = - \frac{1}{2} \Rightarrow x = \pm 120 °

\Rightarrow u = \frac{\sqrt{2}}{2} \Rightarrow x = \pm 45 °

\Rightarrow u = - \frac{\sqrt{2}}{2} \Rightarrow x = \pm 135 °

Answered by cortano12 last updated on 21/Oct/23

\cos 2 x + (\cos 3 x + \cos x) = 0

\cos 2 x + 2 \cos 2 x \cos x = 0

\cos 2 x (1 + 2 \cos x) = 0

{\begin{cases} \cos 2 x = 0 \Rightarrow x = \pm \frac{π}{4} \\ \cos x = - \frac{1}{2} \Rightarrow x = \pm \frac{2 π}{3} \\ \cos 2 x = \cos \frac{3 π}{2} \Rightarrow x = \pm \frac{3 π}{4} \end{cases}