Question

What is the exact value of \[\tan \frac{ \pi }{ 8 }\] ?

Answer 1

Wouldn't you use the half angle formula or something along those lines?

Answer 2

I honestly have no idea of what I should do to start solving it.

Answer 3

Look up half-angle formulas/

Answer 4

I found this, but I'm still confused. http://mathworld.wolfram.com/Half-AngleFormulas.html

Answer 5

Yes, there you go. Now what you have to do is use the one that involves tan

Answer 6

Okay

Answer 7

And if you look at this, you can see that there are 3 ways to find the half-angle of tan: http://www.wyzant.com/resources/lessons/math/trigonometry/half-angle-double-angle-formulas

Answer 8

tan x/2 = cosecx - cotx
Let pi/4 = x,
cosec pi/4 - cot pi/4 = rt2 - 1

Answer 9

So let's use: \[\LARGE tan\frac{x}{2}=\frac{1-cosx}{sinx}\]
Now, what radian do you know, when divided by 2 gives you \[\LARGE \frac{pi}{8}\]

Answer 10

Let tan(pi/8) = y

and tan2x = 2 tanx / (1 - tan^2 x)

so tan (pi/4) = 2 tan (pi/8) / ( 1 - tan^2 (pi/8))

1 = 2y/(1-y^2)

1-y^2 = 2y

y^2 + 2y - 1 = 0

y = -1 +/- sqrt(2)

since tan (pi/8) is in first quadrant, the value should be positive.

Therefore the answer is -1 + sqrt(2)

Answer 11

Thanks!

Answer 12

exact value till 5 decimal space 0.41421