Find all solutions of the following equation in the interval [0,2π). $2 \cos ^ { 3 } x = 2 \cos x$
A) $0 , \frac { \pi } { 2 } , 2 \pi , \frac { 3 \pi } { 4 }$
B) $0 , \frac { \pi } { 2 } , 3 \pi , \frac { 3 \pi } { 2 }$
C) $0 , \frac { \pi } { 3 } , \pi , \frac { 2 \pi } { 3 }$
D) $0 , \frac { \pi } { 4 } , \pi , \frac { 2 \pi } { 3 }$
E) $0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$

Question

Find all solutions of the following equation in the interval [0,2π).​ $2 \cos ^ { 3 } x = 2 \cos x$ ​
A) $0 , \frac { \pi } { 2 } , 2 \pi , \frac { 3 \pi } { 4 }$
B)​ $0 , \frac { \pi } { 2 } , 3 \pi , \frac { 3 \pi } { 2 }$
C)​ $0 , \frac { \pi } { 3 } , \pi , \frac { 2 \pi } { 3 }$
D)​ $0 , \frac { \pi } { 4 } , \pi , \frac { 2 \pi } { 3 }$
E)​ $0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$

Quizplus · Accepted Answer

The equation $2 \cos ^ { 3 } x = 2 \cos x$ simplifies to $\cos^3 x = \cos x$. Factoring out $\cos x$, we get $\cos x (\cos^2 x - 1) = 0$, which simplifies to $\cos x (\cos x + 1)(\cos x - 1) = 0$. The solutions in the interval [0,2π) for $\cos x = 0$ are $x = \frac{\pi}{2}, \frac{3\pi}{2}$, and for $\cos x = 1$ is $x = 0$, and for $\cos x = -1$ is $x = \pi$.

Find All Solutions of the Following Equation in the Interval 2cos⁡3x=2cos⁡x2 \cos ^ { 3 } x = 2 \cos x2cos3x=2cosx

Find All Solutions of the Following Equation in the Interval $2 \cos ^ { 3 } x = 2 \cos x$