We find α using

`alpha=arctan\ a/b`

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

`R=sqrt(a^2+b^2`

So `R=sqrt(7^2+12^2)=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).

cosine combined

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)

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