Daisy Hatfield

2023-03-26

Find $\mathrm{tan}\left(22.{5}^{\circ }\right)$ using the half-angle formula.

### Answer & Explanation

beredelseyqha

Using the half-angle formula, determine the value.
We know that $\mathrm{tan}\left(2x\right)=\frac{2\mathrm{tan}x}{1-{\left(\mathrm{tan}\left(x\right)\right)}^{2}}$ and
Let $y=\mathrm{tan}\left(x\right)$ where $x={\left(22.5\right)}^{\circ }$
So $2x={\left(45\right)}^{\circ }$
Given that $\mathrm{tan}\left({45}^{\circ }\right)=1$
So $\frac{2y}{1-{y}^{2}}=1$
$⇒1-{y}^{2}=2y⇒{y}^{2}+2y-1=0$
We know if $\alpha$ and $\beta$ are the roots of a quadratic equation of the form $a{x}^{2}+bx+c=0$ then the roots are
$\alpha =\frac{-b+{b}^{2}-4ac}{2a}\beta =\frac{-b-{b}^{2}-4ac}{2a}$
Thus the roots are $2-1$ and $-2-1$
Since, $\mathrm{tan}\left(22.{5}^{\circ }\right)$ lie in first quadrant so we will take the positive value.
Thus, the value of $\mathrm{tan}\left(22.{5}^{\circ }\right)$is $2-1$

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