1. ## 8sin^2(v)-2cos(v)=5

$\displaystyle 8\sin(v)^2-2\cos(v)=5 \Leftrightarrow$
$\displaystyle 8(1-\cos(v)^2)-2\cos(v)=5 \Leftrightarrow$
$\displaystyle 8-8\cos(v)^2-2\cos(v)=5 \Leftrightarrow$
$\displaystyle 1-\cos(v)^2-\frac{1}{4}\cos(v)=\frac{5}{8} \Leftrightarrow$
$\displaystyle \cos(v)^2+\frac{1}{4}\cos(v)-\frac{3}{8}=0 \Leftrightarrow$
$\displaystyle \cos(v)=\frac{-7+-\sqrt{7}}{4}$

The thing is that my book states that I can solve this one partly exact.
That is for half of the inifinite solutions.
But I can get nothing of this, no exact solutions at all.
Where am I wrong?

2. Originally Posted by a4swe
$\displaystyle 8\sin(v)^2-2\cos(v)=5 \Leftrightarrow$
$\displaystyle 8(1-\cos(v)^2)-2\cos(v)=5 \Leftrightarrow$
$\displaystyle 8-8\cos(v)^2-2\cos(v)=5 \Leftrightarrow$
$\displaystyle 1-\cos(v)^2-\frac{1}{4}\cos(v)=\frac{5}{8} \Leftrightarrow$
$\displaystyle \cos(v)^2+\frac{1}{4}\cos(v)-\frac{3}{8}=0 \Leftrightarrow$
$\displaystyle \cos(v)=\frac{-7+-\sqrt{7}}{4}$

The thing is that my book states that I can solve this one partly exact.
That is for half of the inifinite solutions.
But I can get nothing of this, no exact solutions at all.
Where am I wrong?

Seems to me it is like $\displaystyle 8\sin^2(v)$ as opposed to $\displaystyle 8\sin(v^2)$. I think you're worked that out though because your second last line agrees with my working pretty much.

$\displaystyle \cos(v)^2+\frac{1}{4}\cos(v)-\frac{3}{8}=0 \Leftrightarrow$
So, I just factorised here to give:

$\displaystyle (\cos(v)+\frac{3}{4})(\cos(v)-\frac{1}{2})=0$
And then solve

$\displaystyle \cos(v)=\frac{1}{2}$
$\displaystyle v=\frac{pi}{3}$ that's in radians obviously.

That's what I've figured from it but I'm not 100% sure if I'm right but it looks right to me.

3. Originally Posted by Random333
Seems to me it is like $\displaystyle 8\sin^2(v)$ as opposed to $\displaystyle 8\sin(v^2)$. I think you're worked that out though because your second last line agrees with my working pretty much.

$\displaystyle \cos(v)^2+\frac{1}{4}\cos(v)-\frac{3}{8}=0 \Leftrightarrow$
So, I just factorised here to give:

$\displaystyle (\cos(v)+\frac{3}{4})(\cos(v)-\frac{1}{2})=0$
And then solve

$\displaystyle \cos(v)=\frac{1}{2}$
$\displaystyle v=\frac{pi}{3}$ that's in radians obviously.

That's what I've figured from it but I'm not 100% sure if I'm right but it looks right to me.
Looks right to me to.

RonL

4. Originally Posted by Random333
Seems to me it is like $\displaystyle 8\sin^2(v)$ as opposed to $\displaystyle 8\sin(v^2)$. I think you're worked that out though because your second last line agrees with my working pretty much.

Well, yes you are right it's my LaTeX skills that stoped me.
Next time I'll do it right.
Thank you.

5. Speaking of LaTeX, this will make things a little neater:
$\displaystyle \pm$ for your "+-".

-Dan

6. Hello, a4swe!

Am I missing something important?
. . I found all the solutions . . .
And I have no idea what your book means by "exact" solutions.

$\displaystyle 8\sin^2v - 2\cos v \:=\:5$

$\displaystyle 8(1-\cos^2 v)-2\cos v \:=\:5$
$\displaystyle 8-8\cos^2v - 2\cos v \:=\:5$

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$$\displaystyle = \text{ -}\frac{3}{4}\quad\Rightarrow\quad v \:= \:\cos^{-1}\!\left(\text{-}\frac{3}{4}\right) + 2\pi n . . . and those are the exact solutions. 7. Originally Posted by Soroban Hello, a4swe! Am I missing something important? . . I found all the solutions . . . And I have no idea what your book means by "exact" solutions. Why introduce fractions? . . . Don't you hate them as much as I do? We have: .\displaystyle 8\cos^2v + 2\cos v - 3\:=\:0 Factor: .\displaystyle (2\cos v - 1)(4\cos v + 3)\:=\:0 And solve: . . \displaystyle 2\cos v - 1 \:=\:0\quad\Rightarrow\quad \cos v \,= \,\frac{1}{2}\quad\Rightarrow\quad v \,= \,\pm\frac{\pi}{3} + 2\pi n . . \displaystyle 4\cos v + 3 \:=\:0\quad\Rightarrow\quad \cos v$$\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.

You know what it means

RonL