Originally Posted by

**Soroban** Hello, a4swe!

Am I missing something important?

. . I found **all** the solutions . . .

And I have *no idea* what your book means by "exact" solutions.

Why introduce fractions? . . . Don't you *hate* them as much as I do?

We have: .$\displaystyle 8\cos^2v + 2\cos v - 3\:=\:0$

Factor: .$\displaystyle (2\cos v - 1)(4\cos v + 3)\:=\:0$

And solve:

. . $\displaystyle 2\cos v - 1 \:=\:0\quad\Rightarrow\quad \cos v \,= \,\frac{1}{2}\quad\Rightarrow\quad v \,= \,\pm\frac{\pi}{3} + 2\pi n$

. . $\displaystyle 4\cos v + 3 \:=\:0\quad\Rightarrow\quad \cos v$$\displaystyle = \text{ -}\frac{3}{4}\quad\Rightarrow\quad v \:= \:\cos^{-1}\!\left(\text{-}\frac{3}{4}\right) + 2\pi n$

. . . and those *are* the exact solutions.