How do you evaluate sec ((5pi)/12)?

2 Answers
Nghi N.
Apr 4, 2016

2/(sqrt(2 - sqrt3))

Explanation:

sec = 1/cos . Evaluate cos ((5pi)/12)
Trig unit circle, and property of complementary arcs give -->
cos ((5pi)/12) = cos ((6pi)/12 - (pi)/12) = cos (pi/2 - pi/12) = sin (pi/12)
Find sin (pi/12) by using trig identity:
cos 2a = 1 - 2sin^2 a
cos (pi/6) = sqrt3/2 = 1 - 2sin^2 (pi/12)
2sin^2 (pi/12) = 1 - sqrt3/2 = (2 - sqrt3)/2
sin^2 (pi/12) = (2 - sqrt3)/4
sin (pi/12) = (sqrt(2 - sqrt3))/2 --> sin (pi/12) is positive.
Finally,
sec ((5pi)/12) = 2/(sqrt(2 - sqrt3))

You can check the answer by using a calculator.

Bdub
Apr 4, 2016

sec ((5pi)/12)=sqrt6+sqrt2

Explanation:

sec x=1/cosx

sec ((5pi)/12)=1/cos((5pi)/12)

(5pi)/12 = pi/4 +pi/6-> Break up into composite Argument

=1/cos(pi/4 +pi/6)

->use cos (A+B)=cosAcosB-sinAsinB

=1/(cos(pi/4)cos(pi/6)-sin(pi/4)sin(pi/6))

=1/((sqrt2/2)(sqrt3/2)-(sqrt2/2)(-1/2))

=1/(sqrt6/4 -sqrt2/4) = 1/ ((sqrt6-sqrt2)/4)=4/(sqrt6-sqrt2)

=4/(sqrt6-sqrt2) * (sqrt6+sqrt2)/(sqrt6+sqrt2)

=(4(sqrt6+sqrt2))/(6-2) = (4(sqrt6+sqrt2))/4

=sqrt6+sqrt2

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