Chapter
5
Pre-Calculus Assignment Guide
Chapter five examines the relationships that all of the six trigonometric functions have. We will use
trigonometric identities to simplify expressions and solve equations. The chapter ends with some of the trig
formulas that will be helpful to evaluate trig expressions exactly instead of getting approximations from our
calculators. These formulas will be used heavily next year in Calculus. As usual, please do not put the trig
formulas in your short-term memory! Please ask questions regularly in class or stop by to see me or go to the
Math Resource Center in room C117 for extra help.
1.
5.1
Using Fundamental Identities
Pg. 345-346
# 1-37 odd
2.
5.1
Pg 346-347
3.
5.2
Verifying Trigonometric Identities
Pg. 353-354
# 3-10, 21, 22, 23, 25, 29
4.
5.2
Pg. 354-355
5.
5.3
Solving Trigonometric Equations
Pg. 364
# 3, 5, 7-16 all (remember to write a general solution!)
6.
5.3
Pg. 364-365
7.
5.4
Sum and Difference Formulas
Pg. 372-373
# 9, 12, 17, 23, 24, 25, 28, 35, 36, 39, 41, 47, 49
8.
5.5
Double and Half Angle Formulas
Pg. 382
# 10, 13, 19, 21, 23, 25, 33, 37, 41, 47, 49
Jan
7-11
#1
#2
#3
Graphing Review
Due
EC Review Due
Jan
14-18
Review Sheet
Read through Chapter Summary for sections 5.1-5.5. What did you learn?
Review #2 Due
FINAL EXAMS
1,2,5
FINAL EXAMS
9,3,4
FINAL EXAMS
8,7,6
NO SCHOOL
Jan
21-25
10.
# 17-33 odd, 39, 40, 41, 43, 46, 60, 63
MLK DAY
NO SCHOOL
#4
#5
5.1-5.2 Quiz
#6
#7
Jan
28-1
Pg. 386
Review Sheet
# 31, 33, 37, 40, 43, 45, 47, 51, 55, 57
#8
Review
#9
5.3-5.5 Quiz
#10 odds
Review
#10 evens
Review
Ch 5 Test Part I
Feb
4-8
9.
# 40, 43, 47-71 odd, 81, 83, 89, 92
Ch 5 Test
Part II
Even Answers to Chapter 5
Section 5-1 Pg: 346
40. Proof
92. ln csc x sec x
Section 5-2 Pg: 353
(All evens are proofs)
Review for quiz on 5.3-5.5:
Section 5.3 –

5
 n ,
 n
1.
6
6
2 4
2.
,
,
3 3
 13 5 17
3.
,
,
,
12 12 12 12
Section 5.4 –
Section 5-2 Pg: 354
40. proof
Section 5-3 Pg: 364
5
7
8. x   2 n,
 2 n
4
4
10. x 
12. x 
3
 n
4

3
14. x 
  n,

2
 n or 2n
5
  n,
6
16.

2
x    n,
  n,
3
3
x

2
n
3
6
2 6
4
2. proof
3. proof
Section 5.5 –
1.
,
12
5
1
2 3
2
3. proof
2.
n
5 n

or

40. x 
12 2
12
2
3
 3n
46. x 
4
60. x  1.998
Chapter Review Sheet
1. sin 2 
2. 0,  ,

3.
Section 5-4 Pg: 372
12. 2  3
24. sin (190°)
28. cos (0.54)
56
65
Section 5-5 Pg: 382
 3 7 11
, ,
,
2 2 6 6
 5
3
,
3
  7 5
, ,
,
12 3 12 6
4. sin 2 m  sin 2 n
5. sin
6.
10. x 
13  2 13
26
  n,
Section 5-3 Pg: 364
36.
, (2  3 )
1.
4
45
A  n or A 
7. sin 
8.

3
2 3
1
2 2
2
10
10. 
10
12 15  5 210
11a.
195
12 210  5 15
11b.
195
9.
 n
Name:____________________________________
H-PreCalculus
Chapter 5 Review (HW #9)
1.
Verify: cos  sec  cos   sin 2 
2.
Solve: sin  x  sec  x   2sin  x   0 , 0  x  2
3.
Solve: tan  4 x   3 , 0  x  
4.
Verify: sin  m  n  sin  m  n   sin 2 m  sin 2 n
5.
Simplify:
6.
Solve: tan 2 A  3 tan A
sin  5  cos  9   cos  5  sin  9 
7.
Simplify: cos   2 
8.
Find the exact value:
tan 15o
9.
Find the exact value:
sin 112.5o
10.
If sin u 
11.
5
If sin v  13
,   v  32 and cosu  1515 ,
3
5
 


and   u  32 , find cos  u2 
a.
cos(u+v)
b.
sin(u-v)

2
 u   , find:
Review for Chapter 5 Test (HW #10)
Use the half, double, sum and difference formulas to solve these problems.
sin  5  cos  9   cos  5  sin  9 
1) Simplify
2) Find
sin112.5 exactly.
3) Find the exact value of
cos105
 
4) Find the exact value of tan 15
o
5) Find the exact values of cos 2 and tan 2 given tan  
6) Find the exact values of sin

2
and cos

2
given sin  
7) If
5
sin v  13
,   v  32 and cosu  1515 ,
8) If
sin u 
3
5
and
  u  32 ,
find

2

4
and 0   
5
2
 u   , find: the exact values of cos(u+v) and sin(u-v)
cos  u2 
9) Find the exact value of tan of (u+v) if
tan u   34 &
3
2
 u  2 ,&cos v   12
13 &
Use all of our trig formulas to verify these identities..
10) Verify sin 4 A  4 sin A cos A(1  2 sin 2 A)
11) Verify cos 3x  4 cos3 x  3 cos x
12) Simplify:
13) Verify
cos   2 
sin  m  n  sin  m  n   sin 2 m  sin 2 n
14) Verify:
cos  sec  cos   sin 2 
15) Verify:
cos  A  B   cos  A  B   2cos A cos B
Solve these trigonometric equations.
16) Solve for all values of x:
sin  x  sec  x   2sin  x   0
2 cos2 x  cos x  1
17) Solve for all values of x:
18) Solve for all values of x: tan A 
2
3 tan A
19) Solve for all values of x: csc x  cot x  1
2
20) Solve for x on [ 0,2 ) :
tan  4 x   3
21) Solve for x on [ 0,2 ) : cos
x
2

2
2
22) Solve for x on [ 0,2 ) : sin 3 x  
3
3
and    
4
2
3
2

2
v
Answers:
1. sin( 445 )
2.
2 2
2
3.
 2 3
2
5.
2 3
cos 2  257 tan 2 
6.
sin 2 
7.
sin(u  v)  12
4.
8.
9.
10.
11.
12.
13.
14.
15.
16.
5
5
24
7
cos 2  2 5 5
210 5 15
195
cos(u  v)  12
 10
10
56
33
See in class
See in class
sin( )
See in class
See in class
See in class
x  n , 3  2n , 53  2n
17.
x  3  23 n
18.
A  n , 3  n
19.
x  2  n , 34  n
20.
x  12 , 3 , 712 , 56 , 1312 , 43 , 1912 , 116
21.
x  2
22.
x
4
9
, 109 , 169 , 59 , 119 , 179
15 5 210
195
H-PreCalculus
Chapter 5
Targets
Section 5.1:
1.
I can find any exact trig value using the reciprocal, Pythagorean, complementary and even/odd identities.
a.
Find all six trig functions if sec   32 and tan   0.
Find all six trig functions if cot     5 and sin   
b.
2.
I can simplify trig expressions using the reciprocal, Pythagorean, complementary and even/odd identities.
Simplify:
sin x cos 2 x  sin x
c.
Simplify:
e.


b. Simplify:
sec 2 x 1  sin 2 x
cos2 
1sin 
d. Simplify:
sin x  4
sin x  2
Simplify:
sec2 x tan 2 x  sec2 x
f.
sec3 x  sec2 x  sec x  1
g.
Simplify:
 cot x  csc x  cot x  csc x 
h. Simplify:
sin 4 x  cos 4 x
i.
Simplify:
3
sec x  tan x
j.
1
sec x 1
a.
3.
26
26
2
I can verify simple identities.
a.
Verify:  sec  tan   csc  1  cot 
c.
sin x  cos x
sin x
Verify:

cos x sin x
cos x
 sec x csc x
Simplify:
Simplify:

1
sec x 1
cos x sec xcos2 x  sin 2 x
b. Verify:
csc 2  
d. Verify:
tan   
  csc
Section 5.2:
4.
I can verify trig identities by using the following techniques:
i. working with one side only
ii. combining fractions, factoring, rationalizing radicals and squaring binomials
iii. using fundamental identities
iv. converting all trig expressions into sines and cosines to simplify
Verify each of the following:
tan 2   1 cos 2   1   tan 2 
a.



c.
sec x  tan x 
e.
cot 2 
1csc
g.
tan x  tan y
1 tan x tan y
i.

cos x
1sin x
1sin 
sin 
b.
tan x  cot x  sec x csc x
d.
tan 4 x  tan 2 x sec2 x  tan 2 x
f.
1cos
1 cos

1cos
sin 
h.
1  csc(  )
cos(  )  cot(  )
sin   cos 
cos   sin 

 sec  csc 
sin 
cos 
j.
sin3   cos3 
 1  sin  cos 
sin   cos 
k.
csc4   cot 4   2csc2   1
l.
cot x
csc x  1
m.
sec2 x  cot 2  2  x   1

cot x cot y
cot x cot y 1

 sec 
csc x  1
cot x
Section 5.3:
5.
I can solve trig equations on the interval [0,2 ) and for all values.
Solve each of the following on the interval [0, 2π) and on the interval  ,    :
6.
a.
c.
2 cos x  1  0
sec x  2 tan x  0
b.
d.
4 tan2 x  1  tan2 x
cos x  cot x  0
e.
2 sin2 x  5 sin x  3
f.
cos x  sin x tan x  2
g.
8sin 3 x  4sin 2 x  6sin x  3  0
h.
2sin x cos x  4sin x  cos x  2
I can solve trig equations with a graphing calculator.
a-g.
7.
Solve each of the equations in target #5 above with a graphing calculator.
I can solve trig equations involving multiple angles.
Solve each of the following on the interval [0, 2π) and on the interval  ,    :
a.
2 cos 2 x 
c.
2sin 2  2 x   1
e.
20
b.
cos(2x)  2cos x  1  0
d. tan(4x) = 1
2sin 2  2x  1
Section 5.4:
8.
I can use the sum and difference formulas to find the exact value of a trig function in a given quadrant.
Find the exact value of each of the following.
 
tan 105 
o
a. sin 75
c.
9.
o
b.
cos  712 
d.
sin  1912 
I can use the sum and difference formulas to verify identities.
Verify each of the following:
a.
sin  x  32   cos x
b.
cos3x  4cos3 x  3cos x
c.
sin  x  y  sin  x  y   sin 2 x  sin 2 y
d.
tan  x     tan   x   2 tan x
e.

 1
sin   x  
3
 2
f.
cos  A  B   cos  A  B   2cos A cos B

3 cos x  sin x

Section 5.5:
10. I can use the half angle formulas to find the exact value of a trig function.
Find the exact value of each of the following.

o
a. sin 22.5
11.

b.
cos  58 
c.

tan 105o

I can use the half and double angle formulas to find the exact value of a trig function in a given quadrant.
Find the exact value of sin2u, cos2u and tan2u using the double angle formula.

cos u   73 ,
a. sin u  95 ,
b.
2  u 
c.
tan u   14 ,
3
2
  u  32
 u  2
Find the exact value of sin  u2  , cos  u2  and tan  u2  using the half angle formula.
12.
13.

d.
sin u  23 ,
f.
tan u   ,
2
1
4
 u 
3
2
e.
cos u   52 ,
  u  32
 u  2
I can use the half and double angle formulas to verify identities.
2

a. sec  2   2sec
b.
sec2 
c.
sin 4 x  8cos3 x  sin x  4cos x  sin x
d.
e.
cos 3x   cos3 x  3sin 2 x  cos x
f.
g.
cos2 2 x  4sin 2 x  cos 2 x  1
h.
1  4sin 2 x  cos2 x  cos2  2 x 
cos  u2   
tan u  sin u
2tan u
tan  u2   csc u  cot u
csc sec  2csc  2 
I can use the half and double angle formulas to solve trig equations.
Solve each of the following on the interval [0, 2π) and on the interval  ,    :
a.
sin 2 x  sin x  cos 2 x  cos x  1
b.
cos  2 x   3cos x  1  0
c.
3cos  2 x   5cos x  1
d.
sin  2 x   sin x
Chapter 5 Target Answers
7b.
1a.
1b.
2a.
2b.
2c.
2d.
2e.
2f.
2g.
2h.
2i.
2j.
5a.
cos  
sec 
2
3
sin    3 5
csc  35 5
tan  
cot  
5
2
2 5
5
cos  52626
sec   526
sin  
26
26
tan  
1
5
csc  26
cot   5
7d.
 2n
x  6 , 56
10a.
10b.
x  2
10c.
x  2  2n
11a.
x  76 , 116
x  3 , 53
x 3,

2
3
,
4
3
x  3  n ,
,
2
3
5
3

,6,
5
6
 n ,
5
6  2n , 6  2n

x  6 , 56
x  6  2n , 56  2n
7a.
19
2 6

12
4
x  8 , 98 , 78 , 158
x  8  n , 78  n
11b.
1
2 2
2
1

2 2
2
2  3
20 14
81
31
cos 2u 
81
20 14
tan 2u  
31
sin 2u  
12 10
49
31
cos 2u  
49
12 10
tan 2u  
31
sin
u
3 5

2
6
cos
u
3 5

2
6
u 3 5

2
2
u
7
sin 
2
10
tan
11e.
cos
u
3

2
10
tan
u
7

2
3
sin
u
17  4 17

2
34
11f.
x  6 , 56 , 76 , 116

5h.
6 2
sin 75 
4
7
2 6
cos

12
4
(1  3)2
tan105  
or
2
  32
8
17
15
cos 2u 
17
8
tan 2u  
15
sin 2u  
11d.
x  2 , 32
sin
x  3  2n , 53  2n
5g.
x  16 , 516 , 916 , 1316
8d.
x  76  2n , 116  2n
5f.
x  8 , 38 , 58 , 78 , 98 , 118 , 138 , 158
x  16  n4
8c.
x  6  2n , 56  2n
5e.
 2n
x  8  n4
8b.
x  6  n , 56  n
5d.
7c.
8a.
x  3 , 53
x  3  2n ,
5c.
4
3
x  2  n
2
tan 2 x
5
3
11c.
x  4  n2 , 23  2n ,
7e.
 sin 3 x
1
1  sin 
sin x  2
sec 4 x
tan 2 x(sec x  1)
-1
sin 2 x  cos 2 x
3(sec x  tan x)

5b.
3
2
x  4 , 34 , 54 , 74 , 23 , 43
u
17  4 17

2
34
u
tan  4  17
2
x  0 or
x  0  2 k
 5
x ,
or
3 3

5
x   2 k ,
 2 k
3
3
2 4
x
,
or
3 3
2
4
x
 2 k ,
 2 k
3
3
 5
x  0,  , ,
or
3 3

5
x   k ,  2 k ,
 2 k
3
3
cos
13a.
13b.
13c.
sin 2u 
13d.