Chapter 6 Review
d) cos
Section 6.1
1. Determine the restrictions
a)
sin x
b)
cos x
sec x
c)
sin x
7π
π
7π
π
cos + sin
sin
12
3
12
3
3. Write each expression as a single trigonometric function.
tan x
d)
1 − cos x
2. Determine the restrictions for
π
cot x
a) 2sin
sin x + 1
1 + cos θ
sin θ
=
sin θ
1 − cos θ
6
cos
π
b)
6
c) 1 − 2sin 2 15°
d)
3. Simplify each expression
a)
cot x
csc x
b) cot x sin x c)
1
cot x sec x
1 − tan x
cot x − 1
d)
4. Simplify each expression.
a)
c)
2(csc x − cot x)
sin 2 x
+ sin x csc x
cos2 x
2
b) cot x (sec x − 1)
cos x
d)
sin x cot x
1
1
+
f)
sec2 x csc 2 x
2
2
2
e) tan x cos x
π
π
− sin 2
3
3
π
2 tan
6
π
1 − tan 2
6
cos2
4. Simplify each expression using a sum identity.
a) sin (90° + A)
c) sin (π + A)
b) cos (90° + A)
d) cos (2π + A)
5. Simplify each expression using a difference identity.
2
a) sin (90° − A)
π

c) sin  − A
2

b) sin (270° − A)
 3π
− A
d) cos 
 2

6. Simplify each expression
5. Are the following identities? a) sin x sec x = sec x − 1
2
1
1
+
=1
b)
sec x csc x
2
2
a)
c) cot x + tan x = csc x cot x
6. Simplify each expression
b) cos x + tan x sin x
a)
2sin θ
cos 2θ − 1
c)
2sin θ
sec x sin x
−
sin x cos x
4
b)
cos 3x cos x − sin 3x sin x
d)
sin 3 x
cos2 x − cos2 x
7. Determine the exact value of each expression.
c) sin x + cos x cot x
a) cos
π
7. Verify sin x − cos x = 2 sin x − 1 for x = .
6
4
sin 2θ
2π
b) tan 15° c) sin 105° d) cos
3
5π
6
2
8. Verify sec x + sec x cos x = 1 + sec x. for x =
8. Determine whether each equation is true.
a) cos 80° = cos 75° cos 5° − sin 75° sin 5°
π
.
4
b) cos (−24°) = cos 16° − cos 40° c) tan 70° =
cos x
cos x
π
+
= 2cot 2 x . a) Verify for x =
sec x − 1 sec x + 1
6
b) What are the restrictions of the equation in 0°≤x<360°.
9. Given
10. Algebraically change cos x + sin x = 1into cot x + 1 = csc x
2
2
2
2
9. If ∠A &∠B are both in quadrant I, & sin A =
a) sin 28° cos 35° + cos 28° sin 35°
b) cos 10° cos 7° − sin 10° sin 7°
π
π
π
π
cos + sin sin
12
4
12
4
π
π
π
π
d) sin cos − cos sin
3
4
3
4
a) cos 25° cos 5° − sin 25° sin 5°
b) sin 40° cos 20° + cos 40° sin 20°
π
π
π
π
c) sin cos + cos sin
3
6
3
6
5
13
a) cos (A − B) b) sin (A + B) c) cos 2A d) sin 2A
10. If cos A =
12
13
, and ∠A is in quadrant IV, find the exact value
of sin 2A.
Section 6.3 Extra Practice
1. Simplify each expression
c) cos
2. Simplify and then give an exact value for each expression.
5
& cos B =
evaluate each of the following.
Section 6.2 Extra Practice
1. Write each expression as a single trigonometric function.
3
2 tan 30°
1 − tan 2 70°
a)
sec x
tan x
b)
cot 2 x
csc x − sin x
c)
cot x
1 − sin 2 x
2. Factor and simplify each rational trigonometric expression.
tan x − tan x sin 2 x
sin 2 x + sin x − 6
b)
2
5sin x + 15
cos x
2
2
cos x − 4
sin x tan x − tan x
c)
d)
7cos x − 14
sin x tan x + tan x
a)
3. Prove each identity for all permissible values of x.
2
2
2
2
a) csc x(1 − cos x) = 1 b) (tan x − 1) = sec x − 2 tan x
2
2
sin x + cos x
= cos x
c)
sec x
4. Prove each identity.
b)
a)
sec x sin x
−
= cot x
sin x cos x
sin 2x = 1
1
sin x =
2
x = 60° and 120°
7. A student writes the general solution for sin 2x = 1 over the
π
5π
domain 0 ≤ x < 2π as + 2πn,
+ 2πn; n ∈ I.
4
4
1 + tan x
= tan x
1 + cot x
cot x + tan x
= csc x
sec x
csc x + cot x
= cot x csc x
tan x + sin x
cos x + 1
= cot x
c)
sin x + tan x
5. Prove each identity.
b)
c)
6. Determine and correct the mistake in the following work.
a) What error did the student make?
b) Write the correct general solution for this equation.
a)
sin x + tan x
= tan x
cos x + 1
8. Solve cos x − 2 sin x cos x = 0 over the domain 0 ≤x< 2π.
9.
cos x
1 + sin x
6. Prove each identity.
a)
=
1 − sin x
cos x
1 + cos x
sin x
cos x
cos x
=
+
= 2cot 2 x
b)
c
sin x
1 − cos x
sec x − 1 sec x + 1
Solve (sin x − 1)(tan x − 1) = 0 for all values of x.
10. Solve 2 cos 2x + 1 = 0Give the general solution in degrees.
Chapter 6 Answers
Section 6.1
7. Prove the following a) cos (x + y) cos (x − y) = cos x − sin y
1 + cos 2 x
= cot x c) 1 + sin 2x = (sin x + cos x)2
b)
sin 2 x
2
2
d) sec x =
1 + cos 2 x
c)
x ≠ 2πn; n ∈ I and x ≠
2.
θ ≠ πn; n ∈ I
8. Prove each identity a)sec x − sec x = tan x + tan x
2
b) cos x + cos x tan x = sec x
4. a) 2 b) 1 c) sec2 x d) 1 e) sin x cos x f ) 1 5. a) Yes b) No c) No
6. a) cot x b) sec x c) csc x
2
4
9. Consider the equation
2
4
2
1. a)
x ≠
π
π
+ πn; n ∈ I b) x ≠ πn; n ∈ I and x ≠ + πn; n ∈ I
2
2
π
π
cos2 x
1 + sin x
=
.
1 + 2sin x − 3sin 2 x 1 + 3sin x
7.
x ≠
2
n; n ∈ I
3
1. Solve each equation algebraically over the domain 0 ≤ x<2π.
cos 2x = 0
2
cos x − 2 = cos x
b) 2 cos x − 5 sin x − 5 = 0
=
3. Rewrite each equation in terms of cosine only. Then, solve
algebraically for 0 ≤ x < 2π.
a) cos 2x − 5 cos x = 2 b) cot x + 2 = 0 c) 1 + cos x = 2 sin x
5. Solve tan x + 2 tan x + 1 = 0 over the domain 0 ≤ x < 2π.
1
2
= Left side
2
2
2
= 1+
= 1+
+1
1
 π
cos  
 4
2
2
= Left side


 π   π
 π   π
9. a) LS = cos   ÷  sec   − 1 + cos   ÷  sec   + 1
 6   6 
 6   6


2
2
= −
16 16
1
=
2. Solve each equation over the domain 0°≤x<360°.
4. Solve 2 cos x = 1 over the domain −180° ≤ x ≤ 180°.
9
 π
cos  
 4
1
=
+
 π
 π
cos  
cos  
 4
 4
Section 6.4 Extra Practice
2
 π
Right side = 2 sin 2   – 1
 6
 π
 π
 π
π
8. Left side = sec   + sec   cos   Right side = 1 + sec  
 4
 4
 4
 4
12. Prove the identity. cos 3x + 1 = 4cos x − 3cos x + 1
2
−
= −
11. Prove the following identity. 1 + sin 2x = (sin x + cos x)
a) cos 2x = cos 3x
2
c) cot x = 0
1
=
2
b)
d)
d)
3. a) cos x b) cos x c) sin x d) tan x
 π
 π
Left side = sin 4   – cos4  
 6
 6
1 − cos 2θ
. b) State all restrictions.
sin 2θ
a) sin 2x − cos x = 0
2
c) 2cos x − 1 = 0
+ πn; n ∈ I
2
Show that the equation is true for x = 3.2 radians.
10. a) Prove tan θ =
2
=
3
2
3
2
3
 2

 2

÷
− 1 +
÷
+ 1
 3

 3

2
 2 − 3
 2 + 3
3
÷
+
÷


2
3 
3 


2
=
=
=
3
2
×
3
2 −
3
4−2 3
+
3
+
3
2
×
3
4+2 3
12 + 6 3 + 12 − 6 3
= 6
16 − 12
3
2+ 3
 π
RS = 2 cot 2  
 6
=
2
 π
tan 2  
 6
= 2÷
=6
= LS
1
3
c) Example:
Right side = csc x
9b) x ≠ 0°, 180° 10. cos 2 x + sin 2 x = 1
cos 2 x sin 2 x
1
+
=
sin 2 x
sin 2 x
sin 2 x
cot 2 x + 1 = csc 2 x
Section 6.2
 π
1. a) sin 63° b) cos 17° c) cos − 
 6
3
3
b)
2
2
2. a)
3 d) tan
π
3
Left side =
 π
d) sin  
 12 
π
1
3. a) sin
3
2
c) 1 d)
=
=
4. a) cos A b) −sin A c) −sin A d) cos A
=
5. a) cos A b) −cos A c) cos A d) −sin A
1
2
b) 2 −
3 +1
3 c)
2 2
d) −
56
65
b)
63
c) 7
65
d)
25
24
25
Left side =
2
10. −
=
120
169
1. a)
c)
sin x
cos x + 2
b)
1
2
sin x
c) cos x
d) sin x – 1 3a)
7
2. a) tan x b)
5
(
)
= tan x − 2 tan x + 1
4. a) Example: right side = tan x
sin x + cos x
= 1÷
2
Left side =
1 + tan x
1 + cot x
sec x
1
=1+
cos x
=
= cos x
= Right side
=
sin x
cos x
cos x 

÷ 1 +


sin x 
cos x + sin x
cos x
×
sin x
1
sin x
csc x + cot x
Right side = cot x csc x
tan x + sin x
1 + cos x
sin x + sin x cos x
÷
sin x
sin 2 x
5b) Right side = tan x
Left side =
=
=
1
=
sin x
cos x
2
cos x
sin x cos x
= cot x
= Right side
cos x
sin x (cos x + 1)
cos x
sin 2 x
sin x cos x
×
×
1
cos x + 1
1
cos x + 1
cos x
sin x (cos x + 1)
6. a) Right side =
1 + sin x
cos x
Left side =
1 + cos x
sin x
sin x
1 + cos x
Right side =
×
1 − cos x 1 + cos x
cos x
−
cos x + 1
sin x cos x + sin x
= cot x
= Right side
sin x
sin x cos x
sin x + tan x
b) Left side =
sin x
−
sin x + tan x
= tan x
= Right side
cos x + 1
= ( cos x + 1) ×
=
sec x
=
cos x
sin 2 x
= Left side
sin x
= Right side
Left side =
=
cos x
cos x
sin x + cos x
= tan x
b) Example:
Right side = cot x
=
Left side =
cos 2 x
2
2
c) Right side = cot x
sin 2 x − 2 sin x cos x + cos 2 x
3c) Example:
Right side = cos x
2
1
2
= sin 2 x (sin x)
=1
= Right side
2
Left side =
sin x − 2
Left side = csc 2 x 1 – cos 2 x
3b) Left side = ( tan x − 1) 2
=
cos x + sin x
1 + cos x
cos x
=
×
sin x
sin x(1 + cos x )
6.3 Extra Practice
1
sec x
5a) Example:
3
8. a) true b) false c) true d) false
9. a)
cot x + tan x
= Right side
6. a) cos θ b) cos (4x) c) −sin θ d) −sin θ
7. a) −
sin x
sin x 
 cos x
= cos x 
+

 sin x
cos x 
2π
c) cos 30°
3
b) cos
1
=
sin x(1 + cos x )
=
=
=
2
sin x
1 + cos x
sin x
= Left side
=
cos x
1 − sin x
cos x (1 + sin x )
(1 − sin x)(1 + sin x )
cos x(1 + sin x)
1 − sin x
cos x(1 + sin x)
2
2
cos x
1 + sin x
cos x
= Right side
c)
cos x
Left side =
b)
Right side = sec x
cos x
+
sec x − 1
sec x + 1
1 − cos x
= cos x ÷
2
cos x
=
1 − cos x
+ cos x ÷
cos x
+
1 + cos x
=
cos x
cos x
1 + cos x
Right side = 2 cot x
2
2 cos x
=
2
sin x
2
9Verify for x = 3.2:
Left side =
2 cos x
2
sin x
2
Right side =
= (cos x cos y − sin x sin y )(cos x cos y + sin x sin y )
=
= cos 2 x cos 2 y − sin 2 x sin 2 y
2
= cos 2 x − sin 2 y cos 2 x − sin 2 y + sin 2 y cos 2 x
sin x
=
c) Example:
Left side = 1 + sin 2x
1 + 2 cos x − 1
2
;n ∈ I
2
= sin 2 x + 2sin x cos x + cos 2 x
= 1 + 2sin x cos x
= Left side
= cos x(2 cos 2 x − 1) − 2sin x(sin x cos x) + 1
= Right side
= 2 cos3 x − cos x − 2 sin 2 x cos x + 1
= 2 cos3 x − cos x − 2 cos x (1 − cos 2 x) + 1
= 4 cos3 x − 3cos x + 1
Right side = 4cos 3 x – 3cosx + 1
2
2
1 + cos 2 x
2
1 + 2 cos x − 1
2
= Left side
Section 6.4
1. a)
Left side = sec 2 x
=
2 π 4π
π
5π
,
b) no solution c) , π,
3 3
3
3
3π 7 π
4. −135°, −45°, 45°, 135° 5.
,
4 4
2
cos x
3. a)
= Right side
cos 2 x
8. a)
Example:
Left side = sec4 x – sec 2 x
 1 − cos 2 x 
=
2
2
cos x  cos x 
Right side = tan 4 x + tan 2 x
= tan 2 x(tan 2 x + 1)
=
2
sin x
=
π π 5π 3π
π 3π 5π
7π
π 3π 5π
7π
, ,
,
b) , , , and
c) , , , and
6 2 6 2
4 4 4
4
4 4 4
4
d) π 2. a) 0°, 72°, 144°, 216°, 288° b) 270° c) 90°, 270°
1
1
4
2
= cos 2 x cos x − sin 2 x sin x + 1
sin x
d)
cos x
nπ
= cos(2 x + x ) + 1
2 sin x cos x
cos x
= sin 2 x + 2sin x cos x + cos 2 x
= 1 + 2sin x cos x
=
b) θ ≠
Right side = ( sin x + cos x )
12. Example:
Left side = cos 3x + 1
sin 2 x
= Right side
1
1 − (1 − 2 sin 2 θ)
1 + cos 2 x
= 1 + 2sin x cos x
=
sin 2θ
= Left side
=
Right side = ( sin x + cos x )
≈ 1.14153
1 − cos 2θ
= tan θ
= cos 2 x − sin 2 y
Left side =
1 + sin 3.2
1 + 3sin 3.2
11. Example:
2 sin θ cos θ
Left side = 1 + sin 2 x
sin θ
=
= 1 + 2sin x cos x
cos θ
= cos x(1 − sin y ) − sin y (1 − cos x)
2
cos x
2
cos x
≈ 1.14153
Left side = cos ( x + y ) cos ( x – y )
= Right side
b) Example:
Right side = cot x
cos x (cos 2 x + sin 2 x )
Right side =
1 + 2 sin 3.2 − 3sin 3.2 .
7. a) Example: Right side = cos 2 x − sin 2 y
2
2
cos x
cos x
cos 2 3.2
10. a)
Left side = tan θ
2
cos3 x + cos x sin 2 x
1
2
= Right side
=
=
2
=
Right side =
=
1
= Left side
2
cos x
2 cos x
2
1 − cos x
=
=
Left side = cos x + cos x tan 2 x
sin x 
2

1


cos x  cos x 
2
2
sin x
4
cos x
= Left side
2
6. The error was in dividing 1 by 2.
sin 2 x = 1
2 x = 90°
x = 45°
7. a) The student used 2π rather than π; because the equation is
sin 2x = 1, the period of the function is π.
π
5π
+ πn,
+ πn; n ∈ I
4
4
π π 5π 3π
π
π
8. x = , , ,
9. + 2πn, + πn; n ∈ I
6 2 6 2
2
4
b)
10. 60°, 120°, 240°, 300° Gen Sol: 60° + 180°n, 120° + 180°n;
n∈I