Using appropriate inverse trig expressions what is an equation that defines 3x in the equation cos(3x)=-1?

Question

im not usre exactly how to get this. As well as finding all values for it and the values og x between 0-2pi..

Answer 1

if cos(3x) = -1

then 3x = arccos(-1) = cos^-1(-1) which is the arc cosine (also known as the inverse cosine) function

 

cos^-1( -1 ) = 180º 0' 0"  using a calculator with the answer in degrees
= 180º + k×360º (k=..-1,0,1,..) since cosine is a periodic function repeating every 360º
= -180º, 180º, 540º, ..
= 3.14159265 radians + k×2π (k=..-1,0,1,..) where π = pi and we convert from degrees to radians,

                                                                 with 180º = π radians
= -1π, 1π, 3π, ...

 

so 3x = cos^-1(-1)

or 3x = -1π, 1π, 3π, ... for values of such that 0 <= x <= 2π 

or x = -π/3, π/3, π, 5π/3, 7π/3, ... for x in [0, 2π]

or x = {π/3, π, 5π/3}

Answer 2

If 

 

cos(3x) = -1

 

 

and

 

cos(θ) = -1

 

 

Then

 

θ = 3x

 

 

θ = cos^-1(-1)

 

3x = cos^-1(-1)

 

3x = π

 

x = π/3