Supplementary Problems

These problems are intended for more advanced students. Some of them require concepts from Calculus and Linear Algebra.

1. Use the identity to solve each of the following equations:

     (a) 4x³ - 3x = ½;         (b)         (c) x³ - 3x + 1 = 0.

2. Convert each of the following functions into the form with appropriate constants r and and thus find the maximum and minimum values of without using differentiation. Check using calculus.

     (a)

     (b)

3. Prove the following identities and generalize:

     (a)

     (b)

4. Let Show the following:

     (a)

     (b)

     (c)

     (d)

5. Use the Euler Formula to express

     (a) in terms of

     (b) in terms of

6. This problem assumes familiarity with Calculus.

     (a) Find the Taylor series about 0 of cos(x) and sin(x). Note: It is known that both of these series still converge if the real
     number x is replaced by any complex number z.

     (b) Use the results of part (a) and the Euler Formula to find a series for e^xi.

     (c) Replace x with -i in the result of part (b) and simplify the resulting series. How does the "e" introduced in this text
     compare to the "e" (that is, the base of the natural logarithm) introduced in Calculus?

7. Prove that

8. Show that sin(x + y + z) = sin(x)cos(y)cos(z) + cos(x)sin(y)cos(z) + cos(x)cos(y)sin(z) - sin(x)sin(y)sin(z).

9. Show that cos(x + y + z) = cos(x)cos(y)cos(z) - sin(x)sin(y)cos(z) - sin(x)cos(y)sin(z) - cos(x)sin(y)sin(z).

10. (a) With the calculator in radian mode put 'SIN(X)' on level 2 of the stack and, 'X' on level 1, then press RS

      (b) Leave the result of (a) on the stack but change the mode to degrees and repeat part (a).

      (c) Why do you get different results to parts (a) and (b)?

Problems 11 and 12 assume familiarity with Linear Algebra.

11. Let the linear operator T(v) be defined by let be as in Problem 42 of Exercises for Chapter 2 Sections 3, 4, and 5, let and let Verify that T(v) and T(w) have the same effect on as multiplying A and B by had on that triangle. In this sense, the linear operator T and multiplying by are equivalent operations.

12. Show that the linear operator and multiplying by r > 0, are equivalent processes in the sense given in Problem 11 above.

13. Sketch and identify the locus of all points A in the Argand Plane that satisfy the following equations:

      (a) |A - 3i| + |A + 3i| = 10.

      (b) |A - 3i| - |A + 3i| = 1.