We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
42
1
avatar

Find all possible values of \(\cos(\theta)\) if \(\cos(2\theta) = 2\cos(\theta) .\)

 Oct 28, 2019
 #1
avatar+104933 
+2

cos (2θ)  = 2cos(θ)

 

cosθ*cosθ - sinθ^sinθ  = 2cos(θ)

 

cos^2θ - 2cosθ - sin^2θ  =  0

 

cos^2θ - 2cosθ - (1 - cos^2θ)  = 0

 

2cos^2θ - 2cosθ - 1  =  0

 

Let   cosθ  =  x       ....and we have that....

 

2x^2  - 2x  - 1  = 0

 

Using the quadratic formula  x  =

 

2 ±√[ 4 + 8]

_________    =

      4

 

2  ±√12

_______  =

     4

 

2  ± 2√3

________   =

      4

 

1  ±√3

_____  

    2

 

Only  1  -  √3

        _______    is    a possible solution

              2

 

So

 

cos  θ  =   1  - √3

                ______

                     2

 

arccos  1  -  √3

            _______ =    θ  ≈   111.5° + 2pi * n     or    258.5° + 2pi * n

                 2

 

Where n is an integer

 

 

cool cool cool

 Oct 28, 2019

32 Online Users

avatar
avatar