Question

Solve cos^2(x)-cos(x)=0 for x,

Answer 1

Solution:-

cos^2(x) - cos(x) = 0

cos(x)[ cos(x)−1 ]=0

If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.

cos(x)=0

cos(x)−1=0

Set the first factor equal to 0 and solve.

cos(x)=0

Take the inverse cosine of both sides of the equation to extract x from inside the cosine.

x=arccos(0)

The exact value of arccos(0) is π/2.

x=π/2

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from 2π to find the solution in the fourth quadrant.

x=2π−π/2

Simplify 2π−π/2 = 3π/2

Find the period of cos(x) is 2π.

The period of the cos(x) function is 2π so values will repeat every 2π radians in both directions.

x = π/2 + 2πn, 3π/2 + 2πn, for any integer n

Set the next factor equal to 0.

cos(x)−1=0

cos(x)=1

Take the inverse cosine of both sides of the equation to extract x from inside the cosine.

x=arccos(1)

The exact value of arccos(1) is 0.

x=0

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from 2π to find the solution in the fourth quadrant.

x=2π−0

Subtract 0 from 2π.

x=2π

Find the period of cos(x) is 2π.

The period of the cos(x) function is 2π so values will repeat every 2π radians in both directions.

x=2πn, 2π + 2πn, for any integer n

The final solution is all the values that make cos(x)[ cos(x)−1 ]=0 true.

x = π/2 + πn, 2πn, for any integer n.