Question

Calcule tg 105o e cotg 15o.

Answer 1

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$\boxed{\begin{array}{l}\sf tg(105^\circ)=tg(60^\circ+45^\circ)\\\sf tg(105^\circ)=\dfrac{tg(60^\circ)+tg(45^\circ)}{1-tg(60^\circ)\cdot tg(45^\circ)}\\\sf tg(105^\circ)=\dfrac{\sqrt{3}+1}{1-\sqrt{3}}\\\sf tg(105^\circ)=\dfrac{(\sqrt{3}+1)}{(1-\sqrt{3})}\cdot\dfrac{(1+\sqrt{3})}{(1+\sqrt{3})}\\\sf tg(105^\circ)=\dfrac{\sqrt{3}+3+1+\sqrt{3}}{1+\sqrt{3}-\sqrt{3}-3}\\\sf tg(105^\circ)=\dfrac{4+2\sqrt{3}}{-2}\\\sf tg(105^\circ)=-2-\sqrt{3}\end{array}}$

$\boxed{\begin{array}{l}\sf tg(15^\circ)=tg(45^\circ-30^\circ)\\\sf tg(15^\circ)=\dfrac{tg(45^\circ)-tg(30^\circ)}{1+tg(45^\circ)\cdot tg(30^\circ)}\\\sf tg(15^\circ)=\dfrac{1-\frac{\sqrt{3}}{3}}{1+1\cdot\frac{\sqrt{3}}{3}}\\\sf tg(15^\circ)=\dfrac{\frac{3-\sqrt{3}}{3}}{1+\frac{\sqrt{3}}{3}}\\\sf tg(15^\circ)=\dfrac{\frac{3-\sqrt{3}}{\backslash\!\!\!3}}{\frac{3+\sqrt{3}}{\backslash\!\!\!3}}\\\sf tg(15^\circ)=\dfrac{3-\sqrt{3}}{3+\sqrt{3}}\end{array}}$

$\boxed{\begin{array}{l}\sf tg(15^\circ)=\dfrac{(3-\sqrt{3})}{(3+\sqrt{3})}\cdot\dfrac{(3-\sqrt{3})}{(3-\sqrt{3})}\\\sf tg(15^\circ)=\dfrac{9-2\sqrt{3}+3}{9-3}\\\sf tg(15^\circ)=\dfrac{12-2\sqrt{3}}{6}\\\sf tg(15^\circ)=\dfrac{\backslash\!\!\!2\cdot(6-\sqrt{3})}{\backslash\!\!\!2\cdot3}\\\sf tg(15^\circ)=\dfrac{6-\sqrt{3}}{3}\\\sf cotg(15^\circ)=\dfrac{3}{6-\sqrt{3}}\\\sf cotg(15^\circ)=\dfrac{3}{(6-\sqrt{3})}\cdot\dfrac{(6+\sqrt{3})}{(6+\sqrt{3})}\end{array}}$

$\large\boxed{\begin{array}{l}\sf cotg(15^\circ)=\dfrac{3(6+\sqrt{3})}{6^2-\sqrt{3^2}}\\\sf cotg(15^\circ)=\dfrac{3(6+\sqrt{3})}{36-3}\\\sf cotg(15^\circ)=\dfrac{\diagdown\!\!\!\!3(6+\sqrt{3})}{\diagdown\!\!\!\!\!\!33}\\\sf cotg(15^\circ)=\dfrac{6+\sqrt{3}}{11}\end{array}}$