We want to fill in the blank in the following sentence.
The solution of the equation 2 cos x - 1 = 0 is given by π3+2nπ and 5π3+2nπ, where n is an integer.
Consider the given equation.
2cos x - 1 = 0
When solving trigonometric equations, a common first step is to isolate the trigonometric function, as this makes it easier to find the values of the variable that satisfy the equation. Here, we will isolate cos x.
The solutions in the interval [0,2π) are π3 and 5π3. Because cos x has a period of 2π, there are infinitely many solutions which can be represented by the following equations where n is an integer.
π3+2nπ and 5π3+2nπ
These expressions represent the infinite solutions of the provided trigonometric equation. Together, they constitute the general solution for the given equation. We can now complete the sentence.
The general solution of the equation 2 cos x - 1 = 0 is given by π3+2nπ and 5π3+2nπ, where n is an integer.
