Question 1139293


Use a double-angle identity to find the exact value of each expression. 

1.
 {{{sin( 120 )}}}..........use identity {{{sin( 2a) = 2sin(a) cos(a)}}}

={{{sin( 2*60)}}}

={{{2sin(60) cos(60)}}}

={{{2sqrt(3)/2*(1/2)}}}

={{{cross(2)sqrt(3)/2*(1/cross(2))}}}

={{{sqrt(3)/2}}}


2. 

{{{tan (60)}}}° 

={{{tan (2*30) }}}

={{{sin (2*30) /cos(2*30)}}}.........................{{{sin( 2a) = 2sin(a) cos(a)}}} and {{{cos (2a) = cos^2(a) - sin^2(a) }}}

={{{(2sin(30) cos(30))/(cos^2(30) - sin^2(30) )}}}

={{{(2(1/2) (sqrt(3)/2))/(3/4 - 1/4 )}}}

={{{(1* (sqrt(3)/2))/(2/4 )}}}

={{{4sqrt(3)/4}}} 

={{{sqrt(3)}}}



3.

{{{ cos (4pi/3 )}}}

={{{cos (2*(2pi/3 ))}}}

= {{{cos^2(2pi/3) - sin^2(2pi/3) }}}

={{{1/4-3/4}}}

={{{-2/4}}}

={{{-1/2}}}



4. 

{{{sin( 5pi/3 )}}}

={{{sin( 3pi/3 +2pi/3)}}}

={{{sin( pi +2pi/3)}}}..........use {{{sin (a + b) = sin (a) cos( b) + sin (b) cos (a)}}}

={{{sin( pi)cos(2pi/3) +sin(2pi/3)cos(pi)}}}

={{{0*(-1/2) +(sqrt(3)/2)(-1)}}}

={{{-sqrt(3)/2}}}