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

Consider the function We shall show that this is a sinusoidal wave and find that the amplitude is and the 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 to-do: add diagram. We'll first use a geometric argument that actually shows a more general result, that: is a sinusoidal wave. By setting , it will follow that is sinusoidal. 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. We now draw a line of length at angle (where that angle is measured relative to a line parallel to the x axis). The y-coordinate of is the y-coordinate of plus the vertical displacement from to . In other words its y-coordinate is . However, there is another way to look at the y coordinate of point . The line does not change in length as we change - all that happens is that the triangle rotates about O. In particular, rotates about O. Hence, the y-coordinate of is a sinusoidal function (we can see this from the 'unit circle' definition mentioned earlier). The amplitude is the length of and the phase is . Algebraic Argument 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 . 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?