How do you find the exact value of #tan(cos^-1 (1/2))#?

2 Answers
Feb 17, 2017

tan (pi/3) = sqrt3
tan (-pi/3) = - sqrt3

Explanation:

#cos ^-1 (1/2)# --> #arccos (1/2)#
Trig table and unit circle -->
#cos x = 1/2# --> arc #x = +- pi/3#
There are 2 answers for #(0, 2pi)#
#tan (pi/3) = sqrt3#
#tan (-pi/3) = - sqrt3#

Mar 20, 2017

see below

Explanation:

From the #color(red)(cos^-1(1/2) # we have the side #color(red)(adjacent# to #color(red)(color(red)(angle theta=1# and the side #color(red)(hypote n use = 2# . So this is a #color(red)(30^@-60^@-90^@# triangle and #color(red)(theta = 60^@#therefore the #color(red)(opposite# to #color(red)(angle theta = sqrt3#.

Note that from the restrictions for the range of inverse circular functions #color(red)(cos^-1 x# is restricted to quadrants 1 and 2 and since the argument is positive #color(red)(1/2)# our answer will be in quadrant 1 only.

Hence,
#color(blue)(tan(cos^-1(1/2))=tan theta=(opposite)/(adjacent)=sqrt3/1=sqrt3#