Question #1eff0

1 Answer
Mar 30, 2017

see below

Explanation:

Use the Double Argument Property #color(red)(tan 2theta=(2tan theta)/(1-tan^2 theta)#

Since #color(red)(cos theta# is positive it means that #color(red)(theta# is either in quadrant I or IV. I will find both and you can pick the answer according to the right quadrant that is given in the problem.

Here it is,

#color(purple)(Quadrant I#.

If #color(red)(cos theta = 5/13# then #color(red)(adjacent = 5 and hypote n use = 13# therefore by using pythagorean property we have #color(red)(opposite = 12#. Therefore, #color(red)(tan theta=(opposite)/(adjacent)=12/5#

Now put it in to the formula for #color(red)(tan 2 theta#. That is,

#color(red)((2tan theta)/(1-tan^2 theta)=color(blue)((2(12/5))/(1-(12/5)^2#

#=color(blue)((24/5)/(1-144/25)#

#=color(blue)((24/5)/((25-144)/25)#

#=color(blue)((24/5)/(-119/25 #

#=color(blue)(24/5 * -25/119#

#=color(blue)(-120/119#

#color( Purple)(Quadrant IV#

If #color(red)(cos theta = 5/13# then #color(red)(adjacent = 5 and hypote n use = 13# therefore the #color(red)(opposite = -12#. Thus, #color(red)(tan theta=-12/5#

#color(red)((2tan theta)/(1-tan^2 theta)=color(blue)((2(-12/5))/(1-(12/5)^2#

#=color(blue)((-24/5)/(1-144/25)#

#=color(blue)((-24/5)/((25-144)/25)#

#=color(blue)((-24/5)/(-119/25 #

#=color(blue)(24/5 * 25/119#

#=color(blue)(120/119#