Question #b9a38

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Steve M
Aug 2, 2017

# arctan(sqrt(3)) = pi/3 iff tan(pi/3) = sqrt(3) #

Using #tan^2 theta+1=sec^2 theta# and #cos(pi/3) = 1/2# we have:

# tan^2 (pi/3) +1=sec^2 (pi/3)#

# :. tan^2 (pi/3) +1=1/cos^2 (pi/3)#

# :. tan^2 (pi/3) +1=1/(1/2)^2#

# :. tan^2 (pi/3) +1=1/(1/4)#

# :. tan^2 (pi/3) +1=4#

# :. tan^2 (pi/3) =3#

# :. tan (pi/3) = +- sqrt(3) #

But as #pi/3# is acute # => tan (pi/3) = sqrt(3) #

# => arctan (sqrt(3)) = pi/3 \ \ \ # QED

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