Supplementary Problems

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

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. For every complex number z, the following series converges:

Find (with some help from Problem 5) the series for:

(a) cos(z);       (b) sin(z);       (c)        *(d) e^axcos(bx), where a and b are real numbers.
 

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 press RS SYMBOLIC DA m-OK to get into the differentiation dialog box. Enter 'SIN(X)' in the EXPR: box, X in the VAR: box, and make sure that "Symbolic" is chosen in the RESULT: box, then press m-OK.

(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)?