Question: "Find tan22.5∘ using the half-angle formula."

Daisy Hatfield

Answered question

2023-03-26

Find

\tan (22 . 5^{\circ})

using the half-angle formula.

beredelseyqha

Beginner2023-03-27Added 4 answers

Using the half-angle formula, determine the value.
We know that

\tan (2 x) = \frac{2 \tan x}{1 - {(\tan (x))}^{2}}

and
Let

y = \tan (x)

where

x = {(22.5)}^{\circ}

2 x = {(45)}^{\circ}

Given that

\tan (45^{\circ}) = 1

\frac{2 y}{1 - y^{2}} = 1

\Rightarrow 1 - y^{2} = 2 y \Rightarrow y^{2} + 2 y - 1 = 0

We know if

α

and

β

are the roots of a quadratic equation of the form

a x^{2} + b x + c = 0

then the roots are

α = \frac{- b + b^{2} - 4 a c}{2 a} β = \frac{- b - b^{2} - 4 a c}{2 a}

Thus the roots are

2 - 1

and

- 2 - 1

Since,

\tan (22 . 5^{\circ})

lie in first quadrant so we will take the positive value.
Thus, the value of

\tan (22 . 5^{\circ})

2 - 1

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