Question 1158677
If you google "double angle formulas", you can find
the cos, sin, and tan of {{{ 2theta }}}
------------------------------------------
You are given that {{{ cos( theta ) = 3/5 }}}
and {{{ 0 < theta < 90 }}}
This means the angle is in the 1st quadrant where the
cos, sin, and tan are all positive (+) / (+)
--------------------------------------------
(a) 
{{{ sin( 2*theta ) = 2*sin( theta )*cos( theta ) }}} ( look it up online )
Note that if  {{{ cos( theta ) = 3/5 }}}, you have a 3-4-5 triangle, so
{{{ sin( theta ) = 4/5 }}} and {{{ tan( theta ) = 4/3 }}}
{{{ sin( 2*theta ) = 2*sin( theta )*cos( theta ) }}}
{{{ sin( 2*theta ) = 2*( 4/5 )*( 3/5 ) }}} ( all positive in the 1st quadrant )
{{{ sin( 2*theta ) = 24/25 }}}
-----------------------------------
(b) 
{{{ cos( 2*theta ) = 2*cos^2( theta ) - 1 }}} ( online )
{{{ cos( 2*theta ) = 2*( 3/5 )^2 - 1 }}}
{{{ cos( 2*theta ) = 18/25 - 1 }}}
{{{ cos( 2*theta ) = -7/25 }}}
The negative sign tells me that {{{ 2*theta }}} ends up in the
2nd quadrant where the cosine is negative.
-------------------------------
(c) 
{{{ tan( 2*theta ) = ( 2*tan( theta )) / ( 1 - tan^2( theta ) ) }}} ( online )
{{{ tan( 2*theta ) = ( 2*( 4/3 )) / ( 1 - ( 16/9 )) }}} 
{{{ tan( 2*theta ) = ( 8/3 ) / ( -7/9 ) }}}
{{{ tan( 2*theta ) = ( 8/3 )*( -9/7 ) }}}
{{{ tan( 2*theta ) = -24/7 }}}
It's negative because the tan is negative in the 2nd quadrant
Notice also that:
{{{ tan( 2*theta ) = sin( 2*theta ) / cos( 2*theta ) }}}
{{{ -24/7 = ( 24/25 ) / ( -7/25 ) }}}
{{{ -24/27 = -24/7 }}}
Also notice that I never had to find out what {{{ theta }}} was to
solve these. But from the results, I know that {{{ theta }}} MUST be
greater than 45 degrees, in order for {{{ 2*theta }}} to end up 
in the 2nd quadrant. I'll show this using my calculator:
{{{ cos( theta ) = 3/5 }}}
{{{ theta = arc cos( 3/5 ) }}}
{{{ theta = 53.13 }}} degrees
and {{{ 2*theta  = 106.26 }}} which is in the 2nd quadrant
-------------------------------
Hope all this helps