# What is cos(pi/9)?

• Feb 5th 2013, 08:03 AM
barkybark
What is cos(pi/9)?
I have looked and looked and cannot find an answer in fractional form. I have done multiple trial and errors and have gotten close. The decimal form is (.9396926208), I don't have this on a unit circle and have not found a unit circle that includes this information. Please help (Evilgrin)
• Feb 5th 2013, 08:25 AM
emakarov
Re: What is cos(pi/9)?
Quote:

Originally Posted by barkybark
I have looked and looked and cannot find an answer in fractional form.

What do you mean by the fractional form?

Quote:

Originally Posted by barkybark
I have done multiple trial and errors and have gotten close. The decimal form is (.9396926208), I don't have this on a unit circle and have not found a unit circle that includes this information.

What don't (and do) you have on the unit circle?

$\cos(\pi/9)$ is the positive root of the equation $4x^3 - 3x = 1/2$ (why?). Even though it is a real number, I don't know how to express it without radicals and the imaginary unit.
• Feb 5th 2013, 06:20 PM
Prove It
Re: What is cos(pi/9)?
We are looking for the real solution to \displaystyle \begin{align*} 4x^3 - 3x = \frac{1}{2} \end{align*}. A brief look at the graph shows that there are, in fact, three real solutions to this equation.

If we try to solve \displaystyle \begin{align*} 4x^3 - 3x = \frac{1}{2} \end{align*} using the Cubic Formula, first note that this is the same as \displaystyle \begin{align*} x^3 - \frac{3}{4}x = \frac{1}{8} \end{align*}, a depressed cubic (i.e., one without an \displaystyle \begin{align*} x^2 \end{align*} term).

Now we look for two numbers, s and t, so that \displaystyle \begin{align*} 3st = -\frac{3}{4} \end{align*} and \displaystyle \begin{align*} \end{align*}s^3 - t^3 = \frac{1}{8}. It turns out that \displaystyle \begin{align*} x = s - t \end{align*} will be a solution to our cubic equation.

So now we need to find s and t. First note that \displaystyle \begin{align*} s = -\frac{1}{4t} \end{align*}, and substituting gives

\displaystyle \begin{align*} \left( -\frac{1}{4t} \right)^3 - t^3 &= \frac{1}{8} \\ -\frac{1}{64t^3} - t^3 &= \frac{1}{8} \\ -\frac{1}{64} - t^6 &= \frac{t^3}{8} \\ 1 + 64t^6 &= -8t^3 \\ 64t^6 + 8t^3 + 1 &= 0 \\ 64T^2 + 8T + 1 &= 0 \textrm{ if we let } T = t^3 \\ T &= \frac{8 \pm \sqrt{8^2 - 4(64)(1)}}{2(64)} \\ T &= \frac{8 \pm \sqrt{64 - 256}}{128} \\ T &= \frac{ 8 \pm \sqrt{ -192 } }{128} \\ T &= \frac{ 8 \pm 8\sqrt{ 3 }\, i }{128} \\ T &= \frac{1 \pm \sqrt{3}\,i}{16} \\ t^3 &= \frac{1 \pm \sqrt{3}\,i}{16} \\ t &= \sqrt[3]{\frac{1 \pm \sqrt{3}\,i}{16}} \end{align*}

Just taking the first solution, \displaystyle \begin{align*} t = \sqrt[3]{\frac{1 + \sqrt{3}\,i}{16}} = \frac{\sqrt[3]{1 + \sqrt{3}\,i}}{2\sqrt[3]{2}} \end{align*} and remembering that

\displaystyle \begin{align*} s &= -\frac{1}{4t} \\ &= -\frac{1}{4 \left( \frac{\sqrt[3]{1 + \sqrt{3}\,i}}{2\sqrt[3]{2}} \right) } \\ &= -\frac{\sqrt[3]{2}}{2\sqrt[3]{1 + \sqrt{3}\,i}} \end{align*}

which means one of the solutions to the cubic is

\displaystyle \begin{align*} x &= s - t \\ &= -\frac{\sqrt[3]{2}}{2\sqrt[3]{1 + \sqrt{3}\,i}} - \frac{\sqrt[3]{1 + \sqrt{3}\,i}}{2\sqrt[3]{2}} \end{align*}

Surprisingly, this is a real number, so you will be able to long divide your cubic to find a resulting quadratic equation, and then use the Quadratic Formula to find the remaining two roots. One of these three roots will be \displaystyle \begin{align*} \cos{\left( \frac{\pi}{9} \right)} \end{align*}. Alternatively, you could use the other solution \displaystyle \begin{align*} t = \sqrt[3]{\frac{1 - \sqrt{3}\,i}{16}} \end{align*} to get a second root.

PHEW!