Solve: cos^-1 x+cos^-1 2x=60°?

Answer 1

x=1/2.

Steve

Answer 2

take the cosine of both sides,
then,
cos [cos^-1 x + cos^-1 2x] = cos 60 = 1/2
=> 2x^2 - sin (cos^-1 x) . sin (cos^-1 2x) = 1/2
since sin^2 x + cos^2 x =1

2x^2 - V(1-x^2).V(1-4x^2) = 1/2

solve for x

Answer 3

You could solve this visually:

x = 1/2

Answer 4

let cos^-1 x = y and cos^-1 2x = z

x = cos y
sin y = sqrt(1-x^2)

sin z = sqrt( 1- 4x^2)
cos (y+z) = cos y cos z - sin y sin z
= x.2x - sqrt((1-x^2)(1-4x^2))

let x^2 = p

we have 1/2 = 2p - sqrt((1-p)(1-4p))
or 1 = 4p - sqrt((1-p)(1-4p))
or 1-4p = sqrt((1-p)(1- 4p))
square both sides

(1-4p)^2 = (1- p)(1-4p)

1-4p =0 or 1-p = 1

p = 1/4 or 0
x = 1/2 or 0 or -1/2
x = 1/2 is solution and x = 0 and -1/2 is superflous

so solution x = 1/2