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Exercises for Chapter III Section 1
Do NOT use a calculator for any of these problems, give only exact answers. (See Preface)
1. Given that 
(a) Find r, a, b such that 
with the given angle 
(b) Find the other 5 trigonometric functions of 
2. Given that 
find a complex number in both polar and rectangular forms
having 
as its argument and then find the other five trigonometric functions of 
3. Let 
be an angle with 
Find r, a, b such that 
with
the given 
and then find 
and 
4. Let 
acute. Find a complex number in both polar and rectangular forms having 
as its
argument and then find 
and 
5. Use the result of Problem 16 (b), Exercises for Chapter II Sections 3, 4, and 5 and definition (D) to find each of the following:
(a) sin(15o) (b) cos(15o) (c) tan(15o).
6. Use the result of Problem 16 (a), Exercises for Chapter II Sections 3, 4, and 5, the results of Example 3 in Section 5 of Chapter II and definition (D) to find each of the following:
(a) sin(105o); (b) cos(105o); (c) tan(105o);
(d) sin(75o); (e) cos(75o); (f) tan(75o).
7. Let 
be as in Problems 3 and 4 above. Use operations on complex numbers and the results of
Problems 3 and 4 to find the following: [Do not use subtraction formulas, double angle formulas, etc.]
8. Given that 
use square roots of complex numbers
[See Problem 29, Exercises for Chapter II Sections 3, 4, and 5.]
to find all possibilities for:
9. Use operations on complex numbers (not sum and difference formulas) to verify each of the following:
(a) 
(b) 
(c) 
(d) 
10. It is clear from definitions (D)
that there are values of 
for which some of the trigonometric functions are not
defined because the denominator of the defining fraction will be zero.
(a) Characterize all the values of 
for which the cosecant and cotangent are not defined.
(b) Characterize all the values of 
for which the tangent and secant are not defined.
11. Prove the Pythagorean Identity; 
for all angles 
12. Let 
Find all six trigonometric functions of 
in terms of c and s.
13. Let 
Find the six trigonometric functions of 
in terms of c for
[HINT: Use the results of problems 11 and 12.]
14. Let 
Express 
in terms of c and s.
15. Use Problem 14 to express 
in terms of 
and 
for n = 1, 2, 3, 4, 5.
16. Express 
in terms of 
for n = 1, 2, 3, 4, 5.
17. Derive the addition, subtraction, and double angle formulas for the cosine, sine, tangent, and cotangent using complex numbers. (Some of these are in Examples 1, 2, and 3.)
18. (a) Derive the half angle formulas 
and
using square roots of complex numbers.
Explain choice of 
sign.
[See Problem 31, Exercises for Chapter II Sections 3, 4, and 5.]
(b) Use part (a) to show that 
(c) Use Problem 28, Exercises for Chapter II Sections 3, 4, and 5 to show that
19. (a) Complete the following table: [HINT: See
Problem 3, Exercises for Chapter II Sections 3, 4, and 5. and
use 
]
| x |  |
 |
 |
 |
0 |  |
 |
 |
 |
| sin(x) | 0 | 1/2 |
(b) Explain why 
(c) Use parts (a) and (b) to tabulate y = sin(x) for 
(d) Graph y = sin(x) for 
20. (a) Graph y = cos(x) for 
(b) Graph y = tan(x) for 
21. Use the formula csc(x) = 1/sin(x) and Problem 19 to graph y = csc(x) for 
and 
22. Graph:
(a) y = sec(x) for 
(b) y = cot(x) for 
and 
23. Consider a right triangle with 
one of its acute angles. Let hyp be the length of the hypotenuse, adj the
length of the side adjacent to 
and opp the length of the side opposite 
Verify each of the following:
24. Use the results of problems 5 and 23 above to find x in each of the following.
(a) |
(b) |
(c) |
(d) |
(e) |
(f) |
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25. There is concern that a church steeple is too high for a new airport runway which is to be extended near by. An airport official stands at a point 24 feet from the point directly below the top of the steeple. From that location the angle of elevation to the top of the steeple is 75o. How tall is the steeple? [HINT: See problem 24 above.] |
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