Question

Find the quadratic equation whose roots are tan(π/8) & tan(5π/8) ?​

Answer 1

Answer:

The required equation is

${x}^{2} + 2x - 1 = 0$

✌️Hope it will help you.✌️

Answer 2

Answer:

let Ф = π/8

2Ф = π/4  

   tan 2Ф = 1  = 2 tan Ф / [ 1 - tan² Ф ]

  1  - tan² Ф = 2 tan Ф

  tan² Ф + 2 tan Ф - 1 = 0

  tan Ф  =  1/2 [ -2 +- √(4 +4) ]  = -1 +- √2

  tan π/8 is > 0 So,  tan π/8 = √2 - 1

The half-angle formula for tangent is:

tan(a/2) = (sin a / (1 + cos a)) = ((1 - cos a) / sin a)

Now we can plug in values:

tan(5π/8) = (sin(5π/4) / (1 + cos(5π/4)) = ((1 - cos(5π/4)) / sin(5π/4)

tan(5π/8) = (-√2/2) / (1 + (-√2/2)) = (1 - (-√2/2)) / (-√2/2)

tan(5π/8) = ((-√2/2)) / ((2 - √2)/2) = ((2 + √2)/2) / (-√2/2)

Now we can solve the first half:

(-√2/2)(2 / (2 - √2))

(-√2/2)((4 + 2√2) / 2)

(-√2/2)(2 + √2)

(-2√2 - 2)/2

-√2 - 1

tan(5pi/8) = -√2 - 1