ringearV

## Answered question

2021-08-16

Use double- and half-angle formulas to find the exact value of the expression: $\mathrm{tan}\frac{\pi }{12}$

### Answer & Explanation

lobeflepnoumni

Skilled2021-08-17Added 99 answers

Use the trigonometry identity:
$\mathrm{tan}2\alpha =\frac{2\mathrm{tan}\alpha }{1-{\mathrm{tan}}^{2}\alpha }$
Trigonometry table:
$\mathrm{tan}2\alpha =\mathrm{tan}\left(\frac{\pi }{6}\right)=\frac{1}{\sqrt{3}}$
call $\mathrm{tan}\left(\frac{\pi }{12}\right)=t$, we get:
$\frac{1}{\sqrt{3}}=\frac{2t}{1-{t}^{2}}$
cross multiply
${t}^{2}+2\sqrt{3}t-1=0$
Solve this quadratic equation for t
$D={d}^{2}-4ac=4\left(3\right)+4=16⇒d=±4$
Therefore two real roots:
$t=\mathrm{tan}\left(\frac{\pi }{12}\right)=-\frac{b}{2a}±\frac{d}{2a}=-\frac{2\sqrt{3}}{2}±\frac{4}{2}=-\sqrt{3}±2$
Since $\mathrm{tan}\left(\frac{\pi }{12}\right)$ is positive, therefore
$\mathrm{tan}\left(\frac{\pi }{12}\right)=2-\sqrt{3}$
Check by calculator
$\mathrm{tan}\left(\frac{\pi }{12}\right)={\mathrm{tan}15}^{\circ }=0.27$
$2-\sqrt{3}=2-1.73=0.27$

Jeffrey Jordon

Expert2021-12-10Added 2605 answers

Answer is given below (on video)

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