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- anonymous

Find the exact value
tan(9pi/8) Using half life formula

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- anonymous

Find the exact value
tan(9pi/8) Using half life formula

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- anonymous

Would the 9pi/8 change to 9pi/4?

- anonymous

Hehe, I like how you called it half-life formula :P
Well, the idea is that
\[\tan(x/2) = \frac{ 1-cosx }{ sinx }\]
What you want to do is rewrite 9pi/8 as x/2. So if 9pi/8 = x/2, what is x?

- anonymous

Oh you already answered that, yes, you would use 9pi/4 in the formula, lol.

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- anonymous

I meant half angle lol, my formula is different from that one

- anonymous

\[\tan \theta/2=\sqrt{1-\cos \theta/1+\cos \theta}\]

- anonymous

It's an equivalent formula. There are 3 formulas you can usefor tan(x/2), i just chose one of the simple ones. These are the 3 you can use:
tan(x/2) =
\[\sqrt{\frac{ 1-cosx }{ 1+cosx }}\]
\[\frac{ 1-cosx }{ sinx }\]
or
\[\frac{ sinx }{ 1+cosx }\]

- anonymous

Oh okay

- anonymous

Yep. So choose whichever one you like and plug in 9pi/4 :)

- anonymous

But 9pi/4 isnt on my unit circle

- anonymous

Well, any value on the unit circle is equivalent to adding or subtracting multiples of 2pi. So what we can do is subtract 2pi from 9pi/4 to get an equivalent angle that wil be on the unit circle.

- anonymous

is it just simply 7pi/4?

- anonymous

Well, 2pi is equivalent to 8pi/4. Subtracting that for 9pi/4, we would have 9pi/4 - 8pi/4 = pi/4.
Does that make sense?

- anonymous

I think so yeah, how did you know what 2pi is equivalent to 8pi/4

- anonymous

\[\frac{ 2\pi }{ 1 }*\frac{ 4 }{ 4 } = \frac{ 8\pi }{ 4 }\]

- anonymous

\[=\sqrt{1-(2/\sqrt2)/1+(2/\sqrt2)}\]
Kay, With my formula I have this

- anonymous

You have the square roots reverse. it should be \(\sqrt{2}/2\) on each of those.

- anonymous

Oh right

- anonymous

Would I then multiple the 2 to everything?

- anonymous

To reduce it, yes :3

- anonymous

Should I leave it like it is after that

- anonymous

That would be up to your professor. You would have:
\[\sqrt{\frac{ 2-\sqrt{2} }{ 2+\sqrt{2} }}\] Now, because of the formula I mentioned, this is equivalent to:
\[\frac{ \sqrt{2} }{ 2+\sqrt{2} }\] which could be rationalized. So considering the two are equal, it really is up to the professor on how much crazy simplification you have to do.

- anonymous

thank you, I have two more problems to do

- anonymous

Thats fine, seems like I have time

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