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

Question

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

anonymous · Answer

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

anonymous · Answer

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 · Answer

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

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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 · Answer

Oh okay

anonymous · Answer

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

anonymous · Answer

But 9pi/4 isnt on my unit circle

anonymous · Answer

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 · Answer

is it just simply 7pi/4?

anonymous · Answer

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 · Answer

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

anonymous · Answer

$$\frac{ 2\pi }{ 1 }*\frac{ 4 }{ 4 } = \frac{ 8\pi }{ 4 }$$

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

Oh right

anonymous · Answer

Would I then multiple the 2 to everything?

anonymous · Answer

To reduce it, yes :3

anonymous · Answer

Should I leave it like it is after that

anonymous · Answer

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 · Answer

thank you, I have two more problems to do

anonymous · Answer

Thats fine, seems like I have time