Question

evaluate sin65/cos75​

Answer 1

$\mathrm{Simplify}$

Answer:

$\frac{\sin \left(65^{\circ \:}\right)}{\cos \left(75^{\circ \:}\right)}=2\left(2+\sqrt{3}\right)\sqrt{2-\sqrt{3}}\sin \left(65^{\circ \:}\right)\quad \begin{pmatrix}\mathrm{Decimal:}&3.50170\dots \end{pmatrix}$

Step-by-step explanation:

$\frac{\sin \left(65^{\circ \:}\right)}{\cos \left(75^{\circ \:}\right)}$

$\mathrm{Use\:the\:following\:identity}:\quad \cos \left(x\right)=\sin \left(90^{\circ \:}-x\right)$

$\cos \left(75^{\circ \:}\right)=\sin \left(90^{\circ \:}-75^{\circ \:}\right)$

$=\frac{\sin \left(65^{\circ \:}\right)}{\sin \left(90^{\circ \:}-75^{\circ \:}\right)}$

$\mathrm{Simplify}$

$=\frac{\sin \left(65^{\circ \:}\right)}{\sin \left(15^{\circ \:}\right)}$

$\sin \left(15^{\circ \:}\right)=\frac{\sqrt{2-\sqrt{3}}}{2}\\\sin \left(15^{\circ \:}\right)$

$\mathrm{Write}\:\sin \left(15^{\circ \:}\right)\:\mathrm{as}\:\sin \left(\frac{30^{\circ \:}}{2}\right)\\\\=\sin \left(\frac{30^{\circ \:}}{2}\right)\\\\\mathrm{Using\:the\:half\:angle\:identity}:\quad \sin \left(\frac{x}{2}\right)=\sqrt{\frac{1-\cos \left(x\right)}{2}}\\\\=\sqrt{\frac{1-\cos \left(30^{\circ \:}\right)}{2}}$$\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(30^{\circ \:}\right)=\frac{\sqrt{3}}{2}\\\\=\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}\\\\\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}=\frac{\sqrt{2-\sqrt{3}}}{2}\\\\=\frac{\sqrt{2-\sqrt{3}}}{2}\\\\=\frac{\sin \left(65^{\circ \:}\right)}{\frac{\sqrt{2-\sqrt{3}}}{2}}\\\\\mathrm{Simplify}\:\frac{\sin \left(65^{\circ \:}\right)}{\frac{\sqrt{2-\sqrt{3}}}{2}}:\quad 2\left(2+\sqrt{3}\right)\sqrt{2-\sqrt{3}}\sin \left(65^{\circ \:}\right)$

$=2\left(2+\sqrt{3}\right)\sqrt{2-\sqrt{3}}\sin \left(65^{\circ \:}\right)$

Answer 2

Answer:

decimal form

3.50170439

$2\left(2+\sqrt{3}\right)\sqrt{2-\sqrt{3}}\sin \left(65^{\circ \:}\right)$