Question

chapter 6 quiz
170 points possible answered: 0/17
question 1
solve 4 cos(4x) = 2 for the smallest positive solution.
give your answer accurate to at least two decimal places.
add work

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Explanation:

Step1: Isolate the cosine function

Divide both sides of the equation \(4\cos(4x) = 2\) by \(4\).
\(\cos(4x)=\frac{2}{4}=\frac{1}{2}\)

Step2: Solve for \(4x\)

We know that if \(\cos\theta=\frac{1}{2}\), then \(\theta = 2n\pi\pm\frac{\pi}{3}\), \(n\in\mathbb{Z}\). For the smallest positive solution, we take \(n = 0\) and the positive case \(\theta=4x=\frac{\pi}{3}\) (since \(\cos\frac{\pi}{3}=\frac{1}{2}\) and we want the smallest positive input to the cosine function).

Step3: Solve for \(x\)

Divide both sides of \(4x=\frac{\pi}{3}\) by \(4\).
\(x=\frac{\pi}{12}\)
Now, calculate \(\frac{\pi}{12}\approx\frac{3.14159}{12}\approx0.26\)

Answer:

\(0.26\)