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### The Pythagorean Identity for Cosine

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Date: 11/11/95 at 3:24:33
From: Mustafa KARA
Subject: Hello from Turkey

Dear sir,

First of all, I want to say to you Thank You for helping me to
solve my math problems. My question is this.

If Sin 9 = x , what is the solution and result of
Cos9.cos18.cos36?

Thank you very much. Good bye.
Mustafa Kara
```

```
Date: 11/12/95 at 14:15:51
From: Doctor Ken
Subject: Re: Hello from Turkey

Hello!

The main things you'll have to know to do this problem are the
formula Sin[t]^2 + Cos[t]^2 = 1, and the formulas for Cos[2t] and
Sin[2t].

For instance, where you have Cos[9], you want to write that in
terms of Sin[9], so solve the "Pythagorean" identity for Cos[t]:

Cos[t] = Sqrt{1 - Sin[t]^2}.  So that means Cos[9] = Sqrt{1 -
x^2}, right?

So you can put that in the first part of your expression, and then
you have:

Cos[9]*Cos[18]*Cos[36] = Sqrt{1 - x^2}*Cos[18]*Cos[36].

You can kind of keep going in this fashion until you get something
that's all in terms of x.  Good luck!

-Doctor Ken,  The Geometry Forum

```
Associated Topics:
High School Trigonometry

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