Find the value of tan π/8.

Question

Find the value of tan \(\frac{\pi}{8}\).

Answer 1

Method 1:

\(\frac{\pi}{8}\) = \(\frac{180°}{8}=\frac{45°}{2}=22\frac{1}{2}\)

We know that tan 2A = \(\frac{2tan\,A}{1-tan^2A}\)

Put A = 22\(\frac{1}{2}\) in the above formula

We get 

On cross multiplication we get

Here a = 1, b = 2, c = -1

Since 22\(\frac{1}{2}\) is acute tan 22\(\frac{1}{2}\) is positive tan 22\(\frac{1}{2}\) = tan \(\frac{\pi}{8}\)

= -1 + √2

= √2 – 1

Method 2:

∴ tan² 22\(\frac{1}{2}\) = (√2 - 1)²

Taking square root, tan² 22\(\frac{1}{2}\) = ±(√2 – 1)

But 22\(\frac{1}{2}\) lies in first quadrant, tan 22\(\frac{1}{2}\) is positive.

∴ tan 22\(\frac{1}{2}\) = √2 – 1

Method 3:

Consider tan A = \(\frac{sin2A}{1+cos2A}\)

Put A =  22\(\frac{1}{2}\) 

= tan 22\(\frac{1}{2}\) = √2 – 1