Find all the solutions of the equation in the interval [0, 2π):
cos 4x(cos x - 1) = 0
so cos4x = 0 means 4x = π/2+2kπ or 3π/2+2kπ
That simplifies to x = π/8 + kπ/4
or cosx = 1
so x=0
cos 4x(cos x - 1) = 0
cos 4x = 0 or cosx = 1
the easy one first:
cosx = 1 , so x = 0, 2π
cos 4x = 0 ,
so 4x = π/2 or 4x = 3π/2
x = π/8 or x = 3π/8
but the period of cos 4x = 2π/4 = π/2
so other solutions are π/8 + π/2, π/8 + 2π/2, π/8 + 3π/2, π/8 + 4π/2,
and 3π/8 , 5π/8, 7π/8, 9π/8, 11π/8, 13π/8 , 15π/8, 2π rest are outside of domain
x = 0, π/8, 3π/8, 5π/8, 7π/9, 9π/8, 11π/8, 13π/8, 15π/8, 2π
( in degrees: 0, 22.5, 67.5, 112.5. 157.5, 202.5, 247.5, 292.5, 337.5, 360 )
To find all the solutions of the equation cos 4x(cos x - 1) = 0 in the interval [0, 2π), we can break it down into two separate equations and solve for x.
First, we set cos 4x = 0:
cos 4x = 0
To find the solutions for this equation, we need to find the values of x that make cos 4x equal to zero. In other words, we need to find the values of x where the cosine function equals zero.
The cosine function equals zero at certain specific angles. These angles are π/2, 3π/2, 5π/2, and so on. The general formula for these angles is (2n + 1)π/2, where n is an integer.
So, for cos 4x = 0, we can set up the equation:
4x = (2n + 1)π/2
Now, let's solve for x:
x = (2n + 1)π/8
So, one set of solutions for cos 4x = 0 in the interval [0, 2π) is x = π/8, 3π/8, 5π/8, 7π/8, 9π/8, 11π/8, 13π/8, and 15π/8.
Next, we solve the second part of the equation: cos x - 1 = 0.
cos x - 1 = 0
To solve this equation, we add 1 to both sides:
cos x = 1
The cosine function equals 1 at certain specific angles. These angles are 0, 2π, 4π, and so on. The general formula for these angles is 2nπ, where n is an integer.
So, for cos x = 1, we can set up the equation:
x = 2nπ
Now, let's solve for x in the interval [0, 2π):
0 ≤ x < 2π
Substituting the values of n, we get the solutions:
x = 0, 2π
Combining the solutions from both parts, the complete set of solutions for cos 4x(cos x - 1) = 0 in the interval [0, 2π) is:
x = π/8, 3π/8, 5π/8, 7π/8, 9π/8, 11π/8, 13π/8, 15π/8, 0, 2π.