Use the graph below of the function to approximate the solutions to $4 \cos ( 2 x ) - \cos ( x ) = 0$ in the interval $[ 0,2 \pi )$. Round your answers to one decimal.

A) $0.7,3.0,1.5,4.7$
B) $3.0,1.5,4.7,5.6$
C) $3.0,2.3,4.0,5.6$
D) $0.7,2.3,4.0,5.6$
E) $0.7,1.5,4.0,6.3$

Question

Use the graph below of the function to approximate the solutions to $4 \cos ( 2 x ) - \cos ( x ) = 0$ in the interval $[ 0,2 \pi )$. Round your answers to one decimal.

A) $0.7,3.0,1.5,4.7$ 
B) $3.0,1.5,4.7,5.6$ 
C) $3.0,2.3,4.0,5.6$ 
D) $0.7,2.3,4.0,5.6$ 
E) $0.7,1.5,4.0,6.3$

Quizplus · Accepted Answer

First, we need to find the values of $x$ where $4\cos(2x) = \cos(x)$. Since $\cos(x) = 2\cos^2(x/2)-1$, we can rewrite this equation as $4\cos(2x) = 2\cos^2(x/2) - 1$. Simplifying, we get $2\cos^2(x/2) - 4\cos(2x) - 1 =0$.

Let's look at the graph of $y=2\cos^2(x/2)-4\cos(2x)-1$ on the given interval:

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We can see that the function has four zeros in the given interval, which correspond to the solutions of the original equation.

Using the graph, we can estimate the approximate values of these zeros as:

- $x\approx 0.7$
- $x\approx 2.3$
- $x\approx 4.0$
- $x\approx 5.6$

Rounding these values to one decimal gives us the answer: D.

Use the Graph Below of the Function to Approximate the Solutions