Question

Value of sin105º. cos 105° is ​

Answer 1

Answer:

-1/4

Explanation:

divide and multiply by 2 then use formula and solve

Answer 2

Answer:

$\displaystyle\sin \left(105^{\circ \:}\right)\cos \left(105^{\circ \:}\right)=-\frac{1}{4}\quad \begin{pmatrix}\mathrm{Decimal:}&-0.25\end{pmatrix}$

Explanation:

$\sin \left(105^{\circ \:}\right)\cos \left(105^{\circ \:}\right)$

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

$\displaystyle\gray{\sin \left(105^{\circ \:}\right)\cos \left(105^{\circ \:}\right)=\frac{\sin \left(2\cdot 105^{\circ \:}\right)}{2}}$

$\displaystyle=\frac{\sin \left(2\cdot \:105^{\circ \:}\right)}{2}$

$\gray{\mathrm{Simplify}}$

$\displaystyle=\frac{\sin \left(210^{\circ \:}\right)}{2}$

$\blue{\sin \left(210^{\circ \:}\right)}$

$\gray{\mathrm{Write}\:\sin \left(210^{\circ \:}\right)\:\mathrm{as}\:\sin \left(180^{\circ \:}+30^{\circ \:}\right)}$

$=\sin \left(180^{\circ \:}+30^{\circ \:}\right)$

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

$=\sin \left(180^{\circ \:}\right)\cos \left(30^{\circ \:}\right)+\cos \left(180^{\circ \:}\right)\sin \left(30^{\circ \:}\right)$

$\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \cos \left(180^{\circ \:}\right)=\left(-1\right)}$

$\gray{\mathrm{Use\:the\:following\:trivial\:identity}:\quad \sin \left(180^{\circ \:}\right)=0}$

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

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

$\displaystyle=0\cdot \frac{\sqrt{3}}{2}+\left(-1\right)\frac{1}{2}$

$\gray{\mathrm{Simplify}}$

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

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

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

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

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

$\displaystyle\gray{\frac{\frac{1}{2}}{2}=\frac{1}{2\cdot \:2}}$

$\displaystyle=-\frac{1}{2\cdot \:2}$

$\gray{\mathrm{Multiply\:the\:numbers:}\:2\cdot \:2=4}$

$\displaystyle=-\frac{1}{4}$