Question

solve the equation in degrees for all exact solutions where appropriate. round approximate answers in degrees to the nearest tenth. write answers using the least possible non - negative angle measures
4\cos\theta = 4\cos2\theta
what is the solution set?
\\( \bigcirc \\)a. \\( \\{ 30 ^ { \circ } + 360 ^ { \circ } n, 120 ^ { \circ } + 360 ^ { \circ } n, 270 ^ { \circ } + 360 ^ { \circ } n \\), where n is any integer\\}
\\( \bigcirc \\)b. \\( \\{ 0 ^ { \circ } + 360 ^ { \circ } n, 120 ^ { \circ } + 360 ^ { \circ } n, 240 ^ { \circ } + 360 ^ { \circ } n \\), where n is any integer\\}
\\( \bigcirc \\)c. \\( \\{ 0 ^ { \circ } + 360 ^ { \circ } n, 150 ^ { \circ } + 360 ^ { \circ } n, 240 ^ { \circ } + 360 ^ { \circ } n \\), where n is any integer\\}
\\( \bigcirc \\)d. \\( \\{ 30 ^ { \circ } + 180 ^ { \circ } n, 150 ^ { \circ } + 180 ^ { \circ } n, 270 ^ { \circ } + 180 ^ { \circ } n \\), where n is any integer\\}

Explanation:

Step1: Simplify the equation

Given \(4\cos\theta = 4\cos2\theta\), divide both sides by 4: \(\cos\theta=\cos2\theta\)
Use the double - angle formula \(\cos2\theta = 2\cos^{2}\theta - 1\), so the equation becomes \(\cos\theta=2\cos^{2}\theta - 1\)
Rearrange to get a quadratic equation in terms of \(\cos\theta\): \(2\cos^{2}\theta-\cos\theta - 1 = 0\)

Step2: Solve the quadratic equation

Let \(x = \cos\theta\), then the quadratic equation is \(2x^{2}-x - 1=0\)
Factor the quadratic: \(2x^{2}-2x + x - 1 = 0\), \(2x(x - 1)+1(x - 1)=0\), \((2x + 1)(x - 1)=0\)
Set each factor equal to zero:

• For \(x - 1=0\), we have \(x=\cos\theta = 1\). Then \(\theta=0^{\circ}+360^{\circ}n\), where \(n\in\mathbb{Z}\) (since \(\cos\theta = 1\) when \(\theta = 0^{\circ}+360^{\circ}n\))
• For \(2x + 1=0\), \(x=\cos\theta=-\frac{1}{2}\). Then \(\theta = 120^{\circ}+360^{\circ}n\) or \(\theta=240^{\circ}+360^{\circ}n\) (since \(\cos\theta=-\frac{1}{2}\) when \(\theta = 120^{\circ}+360^{\circ}n\) or \(\theta = 240^{\circ}+360^{\circ}n\))

Answer:

B. \(0^{\circ}+360^{\circ}n,120^{\circ}+360^{\circ}n,240^{\circ}+360^{\circ}n\), where \(n\) is any integer