Solving Cosine Equations for More Than One Solution

To solve the equationinvert both sides, obtaining

This is not the only the solution. There are an infinite number of solutions. Typically though, we want solutions in the range 0° – 360°.

We can use the cosine curve below to find the solutions.

Draw the horizontal line y=0.4 (because the equation to solve is cos x =0.4). Where this line crosses the curve, drop a line to the x - axis. The solution is where the line crosses the x – axis.

The cosine curve has symmetry, so thatThis often aids the search for a solution.

The two solutions are x=66.42° and x=293.58° .