How do you solve cos [2(x-pi/3)]=sqrt3/ 2?

2 Answers
Peter
May 7, 2016

x = π/2

Explanation:

Given

cos(2(x- (π/3)))= ((3)^(1/2)/2)

let Ω = (2(x- (π/3)),

then

cos Ω = ((3)^(1/2)/2)

we notice that ((3)^(1/2)/2) is a special angle and hence,

the angle Ω that gives cos Ω = ((3)^(1/2)/2) must be π/3

Therefore, Ω =π/3, equate Ω to solve for x,

(2(x- (π/3)) = π/3

x =(π/3)/2+(π/3)

= π/6 + π/3

=π/2

Nghi N.
May 7, 2016

x = (5pi)/12 + 2kpi
x = pi/4 + 2kpi

Explanation:

cos (2x - (2pi)/3) = sqrt3/2
Trig table and unit circle -->
(2x - (2pi)/3) = +- pi/6
a. 2x - (2pi)/3 = pi/6 --> 2x = pi/6 + (2pi)/3 = (5pi)/6 -> x = (5pi)/12
b. 2x - (2pi)/3 = -pi/6 --> 2x = (2pi)/3 - pi/6 = (3pi)/6 = pi/2 -->
x = pi/4
General answers:
x = (5pi)/12 + 2kpi
x = pi/4 + 2kpi
Check.
x = (5pi)/12 --> cos (2x - (2pi)/3) = cos (6pi/12 - (4pi)/12 =
cos (pi/6) = sqrt3/2. OK
x = pi/4 --> cos (2x - (2pi)/3) = cos (pi/2 - (2pi)/3) =
= cos (-pi/6) = sqrt3/2. OK

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