Question

Sin -480 value with calculations

Answer 1

Answer:

$\displaystyle\sin \left(-480^{\circ \:}\right)=-\frac{\sqrt{3}}{2}\quad \begin{pmatrix}\mathrm{Decimal:}&-0.86602\dots \end{pmatrix}$

Step-by-step explanation:

$\sin \left(-480^{\circ \:}\right)$

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

$\gray{\sin \left(-480^{\circ \:}\right)=-\sin \left(480^{\circ \:}\right)}$

$=-\sin \left(480^{\circ \:}\right)$

$\black{\sin \left(480^{\circ \:}\right)}$

$\gray{\mathrm{Rewrite\:the\:angles\:for}\:\sin \left(480^{\circ \:}\right):}$

$\displaystyle\gray{\sin \left(480^{\circ \:}\right)=\sin \left(\frac{6+2}{3}180^{\circ \:}\right)=\sin \left(\left(\frac{6}{3}+\frac{2}{3}\right)180^{\circ \:}\right)=\sin \left(360^{\circ \:}+\frac{2}{3}180^{\circ \:}\right)}$

$\displaystyle=\sin \left(360^{\circ \:}+\frac{2}{3}180^{\circ \:}\right)$

$\gray{\mathrm{Use\:the\:periodiciity\:of\:}\sin :\quad \sin \left(x+360^{\circ \:}\cdot \:k\right)=\sin \left(x\right)}$

$\displaystyle\gray{\sin \left(360^{\circ \:}+\frac{2}{3}180^{\circ \:}\right)=\sin \left(\frac{2}{3}180^{\circ \:}\right)}$

$\displaystyle=\sin \left(\frac{2}{3}180^{\circ \:}\right)$

$\gray{\mathrm{Simplify}}$

$=\sin \left(120^{\circ \:}\right)$

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

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

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

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

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

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

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

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

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

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

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

$\gray{\mathrm{Simplify}}$

$=\dfrac{\sqrt{3}}{2}$

$=-\dfrac{\sqrt{3}}{2}$

Answer 2

Answer:

√3/2 is the answer.............