Witthout-using-mathematical-tables-or-calculator-find-in-surdform-radicals-the-value-tan22-5-

Question Number 59305 by pete last updated on 07/May/19

Witthout using mathematical tables or

calculator, find, in surdform (radicals)

, the value \tan 22 . 5 °

Answered by ajfour last updated on 07/May/19

\tan 2 θ = \frac{2 \tan θ}{1 - \tan^{2} θ}

let θ = 22.5 °, \tan θ = \tan 22.5 ° = q

\Rightarrow 1 = \frac{2 q}{1 - q^{2}}

\Rightarrow q^{2} + 2 q + 1 = 2

\Rightarrow q = \sqrt{2} - 1 .

Commented by pete last updated on 08/May/19

thanks sir

Answered by malwaan last updated on 08/May/19

t a n \frac{θ}{2} = \pm \sqrt{\frac{1 - c o s θ}{1 + s i n θ}}

t a n 22.5 = t a n \frac{45}{2} = + \sqrt{\frac{1 - c o s 45^{°}}{1 + c o s 45^{°}}}

\sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \frac{\sqrt{2}}{2}}} = \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}}

= \sqrt{\frac{{(2 - \sqrt{2})}^{2}}{4 - 2}} = \frac{2 - \sqrt{2}}{\sqrt{2}}

= \frac{2 \sqrt{2} - 2}{2} = \sqrt{2} - 1

Commented by pete last updated on 11/May/19

Thank you boss

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