In this course we have discussed sine waves (with amplitude, period and phase shift).  For certain simple cases, sums of sines and cosines can also be expressed in this way.

 

1.  (a)  Graph  f(x) = sin(x) – 2·cos(x)  on a calculator.  This graph looks like a sine wave. 

Use the calculator graph to compute the amplitude, period and phase shift for f.

Express  f(x)  in the form  a·sin(b(x – h)),  giving a, b, h  with 3 decimal place accuracy.

(I used radian mode and obtained  a ≈ 2.236,  b ≈ 1  and  h ≈ 1.107.  That is:  f(x) ≈ 2.236·sin(x – 1.107). )

 

(b)  Let’s answer this same question algebraically. 

It is convenient to use different letters here.   To find:  constants  r, b, q  so that there is an identity: 

                                    sin(x) – 2·cos(x)  =  r·sin(b(x – q)).

Since  sin(x)  and  cos(x)  each have period  2p  we expect this  f(x) = sin(x) – 2·cos(x)  to have period  2p  as well.  Therefore  b = 1.

 

This function  r·sin(x – q)  can be expanded using the Sum/Difference Identities (p. 562):

            r·sin(x – q)  =            r·(sin(x)cos(q) – sin(q)cos(x))

                               =  (r·cos(q))·sin(x)  –  (r·sin(q))·cos(x)

To find:  r, q  so that this quantity equals  1·sin(x) – 2·cos(x)  for every value  x. 

Compare coefficients of those two expressions to obtain:

                                     r·cos(q)                                    =  1                                    (**)
                                     r·sin(q)                                    =  2.

How can we find values  r  and  q  satisfying these two equations?  Here’s the trick:

First:  The tangent is sine over cosine, and the “r” terms cancel:

               tan(q)   =   =   =   =   2.               Then we can choose  q = arctan(2).

Second:  Since  sin2(q) + cos2(q) = 1,  compute the sum of the squares of the two terms in equation (**) above, to obtain:

                     r2  =  (r·sin(q))2+ (r·cos(q))2  =  22 + 12 = 5.                     Then we can choose  r = .

Use a calculator to estimate   r ≈ 2.236067977   and  q ≈ 1.107148718. 

Note:  These values match the numerical values found in part (a).

 

 

2.  Use the algebraic technique as in Problem 1(b), to answer the following question.

            What is the maximal value for  g(x)  =  3·sin(x) – 5·cos(x) ?
            Find an x-value which yields this maximal value.

 

(My calculator graph indicates the approximate answer:  max = 5.831,  occurring at  x ≈ –3.682  and at  x ≈ 2.601.  Does this match the exact answers you obtained by algebra?)