# \ #
# \mbox{} #
# \mbox{Let's summarize what we are given. We have:} #
# \mbox{A) } \qquad sin( \theta ) = - 5/13 ; #
# \mbox{B) } \qquad \theta \in \mbox{Quadrant IV} . #
# \ #
# \mbox{We are asked to find:} #
# \qquad \qquad \qquad \qquad sin( 2 \theta ), \qquad cos( 2 \theta ), \qquad tan( 2 \theta ) . #
# \ #
# \mbox{Analysis and Computations:} #
# \mbox{1) Thinking ahead toward the double-angle formulas,} \ \mbox{let's compute} \ cos( \theta ); \mbox{we already have} \ \sin( \theta ) . \mbox{Computations:} #
# \qquad \mbox{a)} \qquad cos^2( \theta ) \ = 1 - \sin^2( \theta ) #
# \qquad \qquad \qquad \qquad \qquad \qquad \ = 1 - (- 5/13 )^2 #
# \qquad \qquad \qquad \qquad \qquad \qquad \ = 1 - 25/13^2 \ =
\ { 13^2 - 25 }/13^2 \ = 144/13^2 \ = (12/13)^2. #
# \qquad \qquad \qquad \quad :. \qquad cos( \theta ) \ = \ \pm \sqrt{ (12/13)^2 } \ = \ \pm 12/13 . #
# \mbox{Since we are given:} \quad \ \theta \in \mbox{Quadrant IV}, \mbox{we have:} \quad \ cos( \theta ) > 0. #
# \qquad \qquad \qquad \quad :. \qquad cos( \theta ) \ = \ +12/13 . #
# \qquad \mbox{b) So we now have:} \qquad cos( \theta ) \ = \ 12/13, \quad \mbox{and} \quad \ sin( \theta ) \ = - 5/13. #
# \mbox{2) Now, using the double-angle formulas, we can compute the} \ \ \mbox{desired quantities relatively easily:} #
# \qquad \mbox{a)} \quad sin( 2 \theta ) = 2 sin( \theta )cos( \theta ) = 2 (- 5/13 ) ( 12/13 ) = - 120/169. #
# \qquad \mbox{b)} \quad cos( 2 \theta ) =
cos^2( \theta ) - sin^2( \theta ) = ( 12/13 )^2 - ( - 5/13 )^2 #
# \qquad \qquad \qquad \quad \qquad \qquad= { 12^2 - 5^2 } /13^2 = 119/169. #
# \qquad \mbox{c)} \quad tan( 2 \theta ) =
sin( 2 \theta ) / cos( 2 \theta ) = { - 120/169 } / { 119/169 } = - 120/119. #
# \mbox{3) So, summing up our results, we have: #
# \qquad \mbox{a)} \quad sin( 2 \theta ) = - 120/169. #
# \qquad \mbox{b)} \quad cos( 2 \theta ) = 119/169. #
# \qquad \mbox{c)} \quad tan( 2 \theta ) = - 120/119. #
# \ #
# \mbox{Remark:} #
# \mbox{We were given that} \quad \ \theta \in \mbox{Quadrant IV.} #
# \mbox{Looking at the results we computed in section (3), we can now} \ \ \mbox{also see that:} \qquad \ 2 \theta \in \mbox{Quadrant IV.} #
