What do you mean by the fractional form?

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

is the positive root of the equation (why?). Even though it is a real number, I don't know how to express it without radicals and the imaginary unit.

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- Feb 5th 2013, 07:03 AM #1

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## 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

- Feb 5th 2013, 07:25 AM #2

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## Re: What is cos(pi/9)?

What do you mean by the fractional form?

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

is the positive root of the equation (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, 05:20 PM #3
## Re: What is cos(pi/9)?

We are looking for the real solution to . A brief look at the graph shows that there are, in fact, three real solutions to this equation.

If we try to solve using the Cubic Formula, first note that this is the same as , a depressed cubic (i.e., one without an term).

Now we look for two numbers, s and t, so that and . It turns out that will be a solution to our cubic equation.

So now we need to find s and t. First note that , and substituting gives

Just taking the first solution, and remembering that

which means one of the solutions to the cubic is

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 . Alternatively, you could use the other solution to get a second root.

PHEW!

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