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