Image The ear-pleasing music results from the combination of all the different sound waves coming out of all the musical instruments. It is the sum of all the waves . . .


The Expression (A sin x + B cos x)

It is not uncommon to see the expression (A sin x + B cos x) in the calculations involving trigonometric functions. To simplify the calculations, we often change the expression (A sin x + B cos x) into its equivalent form C sin (x + Image), in which C is equal to (A2 + B2)1/2, sinImage = B/C, and cosImage = A/C

How can we prove the form (A sin x + B cos x) and C sin (x + Image) are equivalent? Starting with the expression C sin (x + Image) and using the composite-argument properties, we have:

C sin (x + Image) = C (sin x cos Image + cos x sin Image)
= (C cos Image) sin x + (C sin Image) cos x

Comparing the above expression with (A sin x + B cos x), we can conclude that

(A sin x + B cos x) and C sin (x + Image)

are equivalent if we can find the appropriate C and Image so that:

(C cos Image) = A
(C sin Image) = B

Do such a C and Image exist? In this case, A and B are given. Our goal is to find the values of C and Image. Since there are two unknowns and two equations, it is possible to find the solution. Squaring both equations and then adding them together, we obtain:

(C2 cos2Image) = A2
(C2 sin2Image) = B2
(C2 cos2Image) + (C2 sin2Image) = A2 + B2
C2 (cos2Image + sin2Image) = A2 + B2
C2 = A2 + B2
C = (A2 + B2)1/2

The positive square root is chosen simply as a matter of convenience. Substituting C into the equation (C sin Image) = B, we have:

sinImage = B/(A2 + B2)1/2
Likewise,  
cosImage = A/(A2 + B2)1/2

Don't feel frustrated if you cannot, at first glance, prove that the two forms are equivalent. Originally it took years to discover this clever combination of algebra and trigonometry.

  [Experiment] [Exercise]