find:
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Exercises for Chapter 3 Section 1.
Do NOT use your calculator for these exercises.
1. Given that
find:
2. Let 
be an angle with 
Find r, a, b such that 
with the given
and then find 
and 
3. Let 
acute. Find a complex number in both polar and rectangular forms having 
as its argument and then find 
and 
4. 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.
5. Let 
be as in Problems 2 and 3 above. Use operations on complex numbers
and the results of Problems 2 and 3 to find the following: [Do not
use subtraction formulas, double angle formulas, etc.]
6. Given that 
use square roots of complex numbers to find all possibilities for
7. Prove the Pythagorean Identity; 
for all angles 
8. Let 
Find all six trigonometric functions of
in terms of c and s.
9. Let 
Find the six trigonometric functions of
in terms of c for
[HINT: Use 
(see Problem 7) and the results of Problem 8.]
10. Let
Express
in terms of c and s.
11. Use Problem 10 to express 
in terms of 
and 
for
n
= 1, 2, 3, 4, 5.
12. Express 
in terms of 
for n = 1, 2, 3, 4, 5.
13. 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.)
14. (a) Derive the formulas 
and 
using
square roots of complex numbers. Explain choice of 
sign.
(b) Use Problem 28, Exercises for Chapter 2 Sections 4, 5, and 6 to show that
15. (a) Complete the following table: [HINT: See Problem
3, Exercises for Chapter 2 Sections 4, 5, and 6. and use 
]
| x |  |
 |
 |
 |
0 |  |
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| 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 
16. (a) Graph y = cos(x) for 
(b) Graph y = tan(x) for 
17. Use the formula csc(x) = 1/sin(x)
and Problem 15 to graph y = csc(x) for 
and 
(a) y = sec(x) for 
(b) y = cot(x) for
and
19. 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:
20. Use the double angle formula for the cosine and the Pythagorean Identity (see Problem 7) to verify each of the following identities:
(a) 
(b) 
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