14. [-/1 Points] DETAILS MY NOTES SCALCET9 A.D.070. Find all values of \( x \) in the interval \( [0,2 \pi] \) that satisfy the equation. (Enter your answers as a comma-separated list.) \[ 16 \cos (x)+8 \sin (2 x)=0 \] \( x=\square \)

Solve the equation \( 16 \cos(x)+8 \sin(2x)=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(16\cos\left(x\right)+8\sin\left(2x\right)=0\) - step1: Rewrite the expression: \(16\cos\left(x\right)+8\times 2\sin\left(x\right)\cos\left(x\right)=0\) - step2: Simplify: \(16\cos\left(x\right)+16\sin\left(x\right)\cos\left(x\right)=0\) - step3: Factor the expression: \(16\cos\left(x\right)\left(1+\sin\left(x\right)\right)=0\) - step4: Calculate: \(\cos\left(x\right)\left(1+\sin\left(x\right)\right)=0\) - step5: Separate into possible cases: \(\begin{align}&\cos\left(x\right)=0\\&1+\sin\left(x\right)=0\end{align}\) - step6: Solve the equation: \(\begin{align}&x=\frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\\&x=\frac{3\pi }{2}+2k\pi ,k \in \mathbb{Z}\end{align}\) - step7: Find the union: \(x=\frac{\pi }{2}+k\pi ,k \in \mathbb{Z}\) The values of \( x \) in the interval \( [0,2\pi] \) that satisfy the equation \( 16 \cos(x)+8 \sin(2x)=0 \) are \( x=\frac{\pi}{2}+k\pi \) where \( k \) is an integer.