I need help solving this problem algebraically, but I need to learn step by step!!
Thanks so much
cos 2x=cos x
How do I do it?
you should know that
cos 2x = 2cos^2 x - 1
so we have
2cos^2 x - 1 = cosx
2cos^2 x - cosx - 1 = 0
(2cosx + 1)(cosx -1) = 0
cosx = -1/2 or cosx = 1
x = 120° or 240° or x = 0 or 360
in radians
x = 0, 2π/3, 4π/3, 2π
Thanks!
To solve the equation cos(2x) = cos(x) algebraically, follow these steps:
Step 1: Use the double angle identity for cosine:
cos(2x) = 2 * cos^2(x) - 1
Step 2: Rewrite the equation:
2 * cos^2(x) - 1 = cos(x)
Step 3: Move all terms to one side to make it a quadratic equation:
2 * cos^2(x) - cos(x) - 1 = 0
Step 4: Factor the quadratic equation:
(2cos(x) + 1)(cos(x) - 1) = 0
Step 5: Set each factor equal to zero and solve for x:
2cos(x) + 1 = 0 or cos(x) - 1 = 0
Step 6: Solve the first equation:
2cos(x) + 1 = 0
2cos(x) = -1
cos(x) = -1/2
Step 7: Solve the second equation:
cos(x) - 1 = 0
cos(x) = 1
Step 8: Find the solutions for x:
To find the solutions, use the unit circle or the cosine values of special angles.
For cos(x) = -1/2:
The solutions for cos(x) = -1/2 are x = (2π/3) + 2πn, (4π/3) + 2πn, where n is an integer.
For cos(x) = 1:
The solution for cos(x) = 1 is x = 2πn, where n is an integer.
Finally, the solutions for the equation cos(2x) = cos(x) are:
x = (2π/3) + 2πn, (4π/3) + 2πn, 2πn, where n is an integer.