Question

cos 15* =??????????.....?????????????????????????????????? ​

Answer 1

Answer:

$\displaystyle\cos \left(15^{\circ \:}\right)=\frac{\sqrt{2+\sqrt{3}}}{2}\quad \begin{pmatrix}\mathrm{Decimal:}&0.96592\dots \end{pmatrix}$

Step-by-step explanation:

$\cos \left(15^{\circ \:}\right)$

$\displaystyle\gray{\mathrm{Write}\:\cos \left(15^{\circ \:}\right)\:\mathrm{as}\:\cos \left(\frac{30^{\circ \:}}{2}\right)}$

$\displaystyle=\cos \left(\frac{30^{\circ \:}}{2}\right)$

$\displaystyle\gray{\mathrm{Using\:the\:half\:angle\:identity}:\quad \cos \left(\frac{x}{2}\right)=\sqrt{\frac{1+\cos \left(x\right)}{2}}}$

$\displaystyle=\sqrt{\frac{1+\cos \left(30^{\circ \:}\right)}{2}}$

$\displaystyle\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(30^{\circ \:}\right)=\frac{\sqrt{3}}{2}}$

$\displaystyle=\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}$

$\displaystyle\sqrt{\frac{1+\frac{\sqrt{3}}{2}}{2}}$

$\displaystyle\gray{\frac{1+\frac{\sqrt{3}}{2}}{2}=\frac{2+\sqrt{3}}{4}}$

$\displaystyle=\sqrt{\frac{2+\sqrt{3}}{4}}$

$\displaystyle\gray{\mathrm{Apply\:radical\:rule\:}\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},\:\quad \mathrm{\:assuming\:}a\ge 0,\:b\ge 0}$

$\displaystyle=\frac{\sqrt{2+\sqrt{3}}}{\sqrt{4}}$

$\gray{\sqrt{4}=2}$

$\displaystyle=\frac{\sqrt{2+\sqrt{3}}}{2}$