We find α using

α has to be in radians for this example, since we are told 0 ≤ α < π/2.

Since a = 7 and b = 12, we have:

α = arctan (7/12) = 0.528

We find R using

So R = √ (7² + 12²) = 13.892

Therefore we can write:

7 sin θ + 12 cos θ = 13.892 cos (θ − 0.528)

To check our answer, we draw the graphs of both y = 7 sin θ + 12 cos θ and y = 13.892 cos (θ − 0.528). We see that they are exactly the same. (Only one is shown).

We observe that our cosine graph has amplitude 13.892 and it has been shifted to the right by 0.528 radians, which is consistent with the expression we obtained: 13.892 cos (θ − 0.528)