write the expression as the sine, cosine, or tangent of a double angle Then find the exact value of the expression. 2 sin 120 cos 120

$$2sin120cos120\\ =sin(120+120)\\ =sin240\\ =sin(180+60) \qquad (third\;\;quadrant\;\;so\;\;negative)\\ =-sin60\\ =\frac{-\sqrt3}{2}$$

2 sin 120 cos 120    ?

$$\\\sin{(\alpha+\beta)}= \sin{(\alpha)} * \cos{(\beta)} + \cos{(\alpha)} * \sin{(\beta)} \\ \sin{(\alpha+\alpha)}= \sin{(\alpha)} * \cos{(\alpha)} + \cos{(\alpha)} * \sin{(\alpha)}\\\\ \boxed{ \sin{(2*\alpha)}= 2*\sin{(\alpha)} * \cos{(\alpha)}}\\\\ \sin{(2*120\ensurement{^{\circ}} )}= 2\sin{(120\ensurement{^{\circ}} )} \cos{(120\ensurement{^{\circ}} )}= \sin{(240 \ensurement{^{\circ}} ) =-0.86602540378$$

$$\\ \sin{(240 \ensurement{^{\circ}} ) = \sin{(360\ensurement{^{\circ}-120 \ensurement{^{\circ} } ) = -\sin{(120 \ensurement{^{\circ} } ) =-\frac{\sqrt{3} } {2} =-0.86602540378$$