# How do you use linear combinations to solve trigonometric equations?

##### 1 Answer

Substitute

where arctan2 is the two parameter, four quadrant inverse tangent.

#### Explanation:

I've been answering all these old math questions. It's hard to know if anyone reads the answers.

The linear combination of a cosine and a sine of the same angle corresponds to a scaling and a phase shift. Let's explain how that works.

The linear combination of a cosine and a sine of the same angle is an expression of the form:

That looks very much like the sum angle formula for sine or the difference angle formula for cosine:

Indeed, we can take our linear combination and transform it into a scaled version of either of these. Pro Tip: When given a choice, prefer cosine to sine.

Let's multiply by a scale factor

Matching to our linear combination leads us to want to solve for

We've seen this before. It's how we turn turn the polar coordinates

Let's remind ourselves how to do that, perhaps in a bit more detail than usually seen.

Squaring and adding we get

Usually this is written as the regular arctangent, but that's not really right. The regular arctangent only covers two quadrants, and doesn't work on the y axis. This is the two parameter, four quadrant inverse tangent, which returns a valid theta for all real input pairs. The

So now we're assured