Trigonometry/Simplifying a sin(x) + b cos(x)

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Consider the function

We shall show that this is a sinusoidal wave, and find its amplitude and phase.

To make things a little simpler, we shall assume that a and b are both positive numbers. This isn't necessary, and after studying this section you may like to think what would happen if either of a or b is zero or negative.

Geometric ArgumentEdit

to-do: add diagram.

We'll first use a geometric argument that actually shows a more general result, that:


is a sinusoidal wave. Since we can set   the result we are trying for with   follows as a special case.

We use the 'unit circle' definition of sine.   is the y coordinate of a line of length   at angle   to the x axis, from O the origin, to a point A.

If we now draw a line   of length   at angle   (where that angle is measure relative to a line parallel to the x axis), its y coordinate is the sum of the two sines.

However, there is another way to look at the y coordinate of point   . The line   does not change in length as we change   , because the lengths of   and   and the angle between them do not change. All that happens is that the triangle   rotates about O. In particular   rotates about O.

This then brings us back to a 'unit circle' like definition of a sinusoidal function. The amplitude is the length of   and the phase is   .

Algebraic ArgumentEdit

The algebraic argument is essentially an algebraic translation of the insights from the geometric argument. We're also in the special case that  and   . The x's and y's in use in this section are now no longer coordinates. The 'y' is going to play the role of   and the 'x' plays the role of   .

We define the angle y by   .

By considering a right-angled triangle with the short sides of length a and b, you should be able to see that

  and   .
Check this

Check that   as expected.


which is (drum roll) a sine wave of amplitude   and phase y.

Check this

Check each step in the formula.

  • What trig formulae did we use?
The more general case

Can you do the full algebraic version for the more general case:


using the geometric argument as a hint? It is quite a bit harder because   is not a right triangle.

  • What additional trig formulas did you need?