Find all solutions of the equation $2 \cos x - 1 = 0$ .
The answer is $A + B k π$ and $C + D k π$ where $k$ is any integer, $0 < A < C < 2 π$ ,
$\begin{array}{l} A = ◻, B = ◻, D = ◻ \\ C = ◻ \end{array}$

Answer

Expert–verified

Hide Steps

Answer

The values are $A = 1.04719755119660$ , $B = 2$ , $C = 5.23598775598299$ , and $D = 2$ .

Steps

Step 1 ：Given the equation $2 \cos x - 1 = 0$ .

Step 2 ：Rewrite the equation as $\cos x = \frac{1}{2}$ .

Step 3 ：The solutions to this equation are $x = \frac{π}{3} + 2 k π$ and $x = - \frac{π}{3} + 2 k π$ where $k$ is any integer.

Step 4 ：So, $A = \frac{π}{3}$ , $B = 2$ , $C = - \frac{π}{3}$ , and $D = 2$ .

Step 5 ：The solutions to the equation are $x = A + B k π$ and $x = C + D k π$ where $k$ is any integer, \(0

Step 6 ：The values are $A = 1.04719755119660$ , $B = 2$ , $C = 5.23598775598299$ , and $D = 2$ .