i nead to find to the nearest degree all values of theta in the interval 0 x 360 that satisfy the equation 3 cos 2 theta+ sin theta-1=0

Please clarify before I attempt this:

is it 3cos (2Ø) ... or is it 3cos^2 Ø .... ?

To find the values of theta that satisfy the equation 3cos(2θ) + sin(θ) - 1 = 0 in the interval 0 ≤ θ ≤ 360, you can follow these steps:

Step 1: Rearrange the equation
Start by rearranging the equation to isolate the trigonometric terms on one side and the constant terms on the other side.
3cos(2θ) + sin(θ) = 1

Step 2: Simplify the equation
Use trigonometric identities to simplify the equation further. We will use the double-angle identity for cosine and the angle-sum identity for sine.
3(2cos^2(θ) - 1) + sin(θ) = 1
6cos^2(θ) - 3 + sin(θ) = 1
6cos^2(θ) + sin(θ) = 4

Step 3: Solve for θ
To find the values of θ, we will use trial and error or numerical methods. Unfortunately, there is no algebraic way to solve this equation precisely. However, we can use a graphing calculator or computer software to visualize the solutions.

One approach is to plot the graph of the function y = 6cos^2(θ) + sin(θ) - 4 and observe the x-values where the graph intersects the x-axis. These intersections represent the solutions to the equation.

Another approach is to use a numerical method like the Newton-Raphson method to approximate the solutions.

Note: In trigonometric equations, it is common to use radian measures rather than degrees. If you need the solutions in radians, you can convert them later.

Remember to always verify the obtained solutions by substituting them back into the original equation to ensure they satisfy it.