Solve 2cos^2θ/2 = cos2θ for 0º≤θ≤360º. (Hint: Graph each side of the equation and find the points of intersection.)
A) ≈73.8º; 260.4º
B) ≈141.3º; 218.7º
C) ≈225.7º; 315.8º
D) ≈51.7º; 138.2º
Using cos^(x/2) = [1 + cos(x)] / 2
Then
2 * [1 + cos(x)] / 2 = cos(2x) →
1 + cos(x) = cos(2x) →
And using cos(2x) = 2cos^2(x) - 1
1 + cosx = 2cos^2(x) - 1 →
2cos^2(x) - cosx - 2 = 0
let x = cosx
2x^2 -x - 2 = 0 .....using the on-site solver......
$${\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{2}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{17}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{4}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{17}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{4}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.780\: \!776\: \!406\: \!404\: \!415\: \!1}}\\
{\mathtt{x}} = {\mathtt{1.280\: \!776\: \!406\: \!404\: \!415\: \!1}}\\
\end{array} \right\}$$
So, using the cosine inverse, and ignoring the second answer, we have.....
cos-1 [(1 - √17)/4] = x ...... so x = 141.331° and x = 218.668°
Using cos^(x/2) = [1 + cos(x)] / 2
Then
2 * [1 + cos(x)] / 2 = cos(2x) →
1 + cos(x) = cos(2x) →
And using cos(2x) = 2cos^2(x) - 1
1 + cosx = 2cos^2(x) - 1 →
2cos^2(x) - cosx - 2 = 0
let x = cosx
2x^2 -x - 2 = 0 .....using the on-site solver......
$${\mathtt{2}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{2}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{17}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{4}}}}\\
{\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{17}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{4}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.780\: \!776\: \!406\: \!404\: \!415\: \!1}}\\
{\mathtt{x}} = {\mathtt{1.280\: \!776\: \!406\: \!404\: \!415\: \!1}}\\
\end{array} \right\}$$
So, using the cosine inverse, and ignoring the second answer, we have.....
cos-1 [(1 - √17)/4] = x ...... so x = 141.331° and x = 218.668°