Question

solve the equation. give a general formula for all the solutions. list six solutions. cos(2θ) = √3/2. find all the solutions to cos(2θ) = √3/2 in 0,π). θ = π/12, 11π/12 (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any num. write the general formula for all the solutions to cos(2θ) = √3/2 based on the smaller angle. θ = (simplify your answer. use angle measures greater than or equal to 0 and less than π. type an exact an using k as the variable )

Explanation:

Step1: Recall cosine inverse property

If $\cos(2\theta)=\frac{\sqrt{3}}{2}$, then $2\theta = 2k\pi\pm\arccos(\frac{\sqrt{3}}{2})$, where $k\in\mathbb{Z}$. Since $\arccos(\frac{\sqrt{3}}{2})=\frac{\pi}{6}$, we have $2\theta = 2k\pi\pm\frac{\pi}{6}$.

Step2: Solve for $\theta$

Divide both sides of $2\theta = 2k\pi\pm\frac{\pi}{6}$ by 2. We get $\theta=k\pi\pm\frac{\pi}{12}$.

Step3: List six solutions

When $k = 0$, $\theta=\frac{\pi}{12}$; when $k = 0$, $\theta =-\frac{\pi}{12}$ (but we usually consider non - negative angles in the general sense, so we can use the positive form for our list). When $k = 1$, $\theta=\pi+\frac{\pi}{12}=\frac{13\pi}{12}$, $\theta=\pi - \frac{\pi}{12}=\frac{11\pi}{12}$, when $k=- 1$, $\theta=-\pi+\frac{\pi}{12}=-\frac{11\pi}{12}$, $\theta=-\pi-\frac{\pi}{12}=-\frac{13\pi}{12}$. Taking non - negative values, six solutions can be $\theta=\frac{\pi}{12},\frac{11\pi}{12},\frac{13\pi}{12},\frac{23\pi}{12},\frac{25\pi}{12},\frac{35\pi}{12}$.
The general formula for all solutions based on the smaller non - negative angle $\frac{\pi}{12}$ is $\theta = k\pi\pm\frac{\pi}{12},k\in\mathbb{Z}$. If we want non - negative angles less than $\pi$, we can write two separate formulas: $\theta=\frac{\pi}{12}+k\pi$ and $\theta = \frac{11\pi}{12}+k\pi$ for $k = 0,1,2,\cdots$. But if we consider all real solutions and want to base on the smaller positive angle $\frac{\pi}{12}$, the general formula is $\theta=k\pi\pm\frac{\pi}{12},k\in\mathbb{Z}$.

Answer:

General formula: $\theta=k\pi\pm\frac{\pi}{12},k\in\mathbb{Z}$
Six solutions: $\frac{\pi}{12},\frac{11\pi}{12},\frac{13\pi}{12},\frac{23\pi}{12},\frac{25\pi}{12},\frac{35\pi}{12}$