Question

Find the solutions on the interval (0, 2pi): sin(x)/(2)+ cos x -1=0

Answer 1

To solve the equation sin(x)/2 + cos(x) - 1 = 0 for x in the interval (0, 2π), we can transform the equation using a half-angle identity for sine and then find the angles on the unit circle corresponding to the positive and negative solutions for sin(x).

Step-by-step explanation:

We have been asked to find the solutions to the equation sin(x)/2 + cos(x) - 1 = 0 on the interval (0, 2π).

Here is the step-by-step solution:

1. Rewrite the equation as sin(x)/2 = 1 - cos(x).
2. Recognizing that 1 - cos(x) is the formula for half-angle sine, rewrite as sin(x)/2 = sin(²/2)^2.
3. Solve the equation for sin(x) by taking the square root of both sides, keeping in mind both the positive and negative solutions.
4. Find the values of x where sin(x) equals the positive and negative square root of the right-hand side.
5. Since we are interested in the interval (0, 2π), use the unit circle to determine the angles that correspond to the sine values found.

By following these steps, we can identify all x in the interval that satisfy the original equation.