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 Argument[edit]

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): \overline {AB} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): B . The line does not change in length as we change , because the lengths of and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): \overline {AB} 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 Argument[edit]

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?