Question

Sin[2arc cos 12/13]​

Answer 1

Answer:

$\displaystyle\sin \left(2\arccos \left(\frac{12}{13}\right)\right)=\frac{120}{169}\quad \begin{pmatrix}\mathrm{Decimal:}&0.71005\dots \end{pmatrix}$

Step-by-step explanation:

$\displaystyle\sin \left(2\arccos \left(\frac{12}{13}\right)\right)$

$\gray{\mathrm{Use\:the\:following\:identity}:\quad \sin \left(2x\right)=2\cos \left(x\right)\sin \left(x\right)}$

$\displaystyle\gray{\sin \left(2\arccos \left(\frac{12}{13}\right)\right)=2\cos \left(\arccos \left(\frac{12}{13}\right)\right)\sin \left(\arccos \left(\frac{12}{13}\right)\right)}$

$\displaystyle=2\cos \left(\arccos \left(\frac{12}{13}\right)\right)\sin \left(\arccos \left(\frac{12}{13}\right)\right)$

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

$\displaystyle=2\left(\frac{12}{13}\right)\sin \left(\arccos \left(\frac{12}{13}\right)\right)$

$\gray{\mathrm{Use\:the\:following\:identity:}\:\sin \left(\arccos \left(x\right)\right)=\sqrt{1-x^2}}$

$\displaystyle=2\left(\frac{12}{13}\right)\sqrt{1-\left(\frac{12}{13}\right)^2}$

$\displaystyle\blue{2\left(\frac{12}{13}\right)\sqrt{1-\left(\frac{12}{13}\right)^2}}$

$\displaystyle\gray{\mathrm{Remove\:parentheses}:\quad \left(a\right)=a}$

$\displaystyle=2\cdot \frac{12}{13}\sqrt{1-\left(\frac{12}{13}\right)^2}$

$\displaystyle\gray{\sqrt{1-\left(\frac{12}{13}\right)^2}=\frac{5}{13}}$

$\displaystyle=2\cdot \frac{5}{13}\cdot \frac{12}{13}$

$\displaystyle\gray{\mathrm{Multiply\:fractions}:\quad \:a\cdot \frac{b}{c}\cdot \frac{d}{e}=\frac{a\:\cdot \:b\:\cdot \:d}{c\:\cdot \:e}}$

$\displaystyle=\frac{12\cdot \:5\cdot \:2}{13\cdot \:13}$

$\gray{\mathrm{Multiply\:the\:numbers:}\:12\cdot \:5\cdot \:2=120}$

$\displaystyle=\frac{120}{13\cdot \:13}$

$\gray{\mathrm{Multiply\:the\:numbers:}\:13\cdot \:13=169}$

$\displaystyle=\frac{120}{169}$