Question

SOLVE

.
..Sin 120/ 1- cos120= ??
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Answer 1

Answer:

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Answer 2

Answer:

$\green{\frac{\sin \left(120^{\circ \:}\right)}{1-\cos \left(120^{\circ \:}\right)}=\frac{\sqrt{3}}{3}\quad \begin{pmatrix}\mathrm{Decimal:}&0.57735\dots \end{pmatrix}}$

Step-by-step explanation:

$\frac{\sin \left(120^{\circ \:}\right)}{1-\cos \left(120^{\circ \:}\right)}$

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

$\gray{\sin \left(120^{\circ \:}\right)=\cos \left(90^{\circ \:}-120^{\circ \:}\right)}$

$=\frac{\cos \left(90^{\circ \:}-120^{\circ \:}\right)}{1-\cos \left(120^{\circ \:}\right)}$

$\gray{\mathrm{Simplify}}$

$=\frac{\cos \left(-30^{\circ \:}\right)}{1-\cos \left(120^{\circ \:}\right)}$

$\gray{\mathrm{Use\:the\:following\:property:}\:\cos \left(-x\right)=\cos \left(x\right)}$

$\gray{\cos \left(-30^{\circ \:}\right)=\cos \left(30^{\circ \:}\right)}$

$=\frac{\cos \left(30^{\circ \:}\right)}{1-\cos \left(120^{\circ \:}\right)}$

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

$\black{\cos \left(120^{\circ \:}\right)}$

$\gray{\mathrm{Write}\:\cos \left(120^{\circ \:}\right)\:\mathrm{as}\:\cos \left(30^{\circ \:}+90^{\circ \:}\right)}$

$=\cos \left(30^{\circ \:}+90^{\circ \:}\right)$

$\gray{\mathrm{Using\:the\:summation\:identity}:\quad \cos \left(x+y\right)=\cos \left(x\right)\cos \left(y\right)-\sin \left(x\right)\sin \left(y\right)}$

$=\cos \left(30^{\circ \:}\right)\cos \left(90^{\circ \:}\right)-\sin \left(30^{\circ \:}\right)\sin \left(90^{\circ \:}\right)$

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

$\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(30^{\circ \:}\right)=\frac{1}{2}}$

$\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(90^{\circ \:}\right)=0}$

$\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(90^{\circ \:}\right)=1}$

$=\frac{\sqrt{3}}{2}\cdot \:0-\frac{1}{2}\cdot \:1$

$\gray{\mathrm{Simplify}}$

$\gray{=-\frac{1}{2}}$

$\gray{=\frac{\frac{\sqrt{3}}{2}}{1-\left(-\frac{1}{2}\right)}}$

$\black{\frac{\frac{\sqrt{3}}{2}}{1-\left(-\frac{1}{2}\right)}}$

$\gray{\mathrm{Apply\:rule}\:-\left(-a\right)=a}$

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

$\gray{\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{\frac{b}{c}}{a}=\frac{b}{c\:\cdot \:a}}$

$=\frac{\sqrt{3}}{2\left(1+\frac{1}{2}\right)}$

$\gray{\mathrm{Join}\:1+\frac{1}{2}:\quad \frac{3}{2}}$

$=\frac{\sqrt{3}}{2\cdot \frac{3}{2}}$

$\gray{\mathrm{Multiply\:}2\cdot \frac{3}{2}\::\quad 3}$

$=\frac{\sqrt{3}}{3}$