(Answers for Chapter 5: Analytic Trigonometry) A.5.1 CHAPTER 5: Analytic Trigonometry SECTION 5.1: FUNDAMENTAL TRIGONOMETRIC IDENTITIES 1) Left Side Right Side Type of Identity (ID) csc x () 1 sin x () Reciprocal ID tan x () 1 cot x () Reciprocal ID tan x () sin x () cos x () Quotient ID tan π 2 − x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ cot x () Cofunction ID cos x () sin π 2 − x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ Cofunction ID sin − x ( ) − sin x () Even / Odd (Negative-Angle) ID cos − x ( ) cos x () Even / Odd (Negative-Angle) ID tan − x ( ) − tan x () Even / Odd (Negative-Angle) ID sin 2 x () + cos 2 x () 1 Pythagorean ID tan 2 x () + 1 sec 2 x () Pythagorean ID 1 + cot 2 x () csc 2 x () Pythagorean ID 2) a) − sec x () ; b) sec 2 θ () ; c) 1; d) csc 4 x () ; e) sin t () ; f) sin α ( ) 3) a) 4 cos θ () ; b) 6 sec θ () ; c) 3tan θ () SECTION 5.2: VERIFYING TRIGONOMETRIC IDENTITIES 1) Solutions will vary.
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(Answers for Chapter 5: Analytic Trigonometry) A.5.1
CHAPTER 5:
Analytic Trigonometry
SECTION 5.1: FUNDAMENTAL TRIGONOMETRIC IDENTITIES
1) Left Side Right Side Type of Identity (ID)
csc x( ) 1
sin x( ) Reciprocal ID
tan x( ) 1cot x( ) Reciprocal ID
tan x( ) sin x( )cos x( ) Quotient ID
tan π
2− x
⎛⎝⎜
⎞⎠⎟
cot x( ) Cofunction ID
cos x( )
sin π2− x⎛
⎝⎜⎞⎠⎟
Cofunction ID
sin − x( ) −sin x( ) Even / Odd (Negative-Angle) ID
cos − x( ) cos x( ) Even / Odd (Negative-Angle) ID
tan − x( ) − tan x( ) Even / Odd (Negative-Angle) ID
sin2 x( ) + cos2 x( ) 1 Pythagorean ID
tan2 x( ) +1 sec2 x( ) Pythagorean ID
1+ cot2 x( ) csc2 x( ) Pythagorean ID
2) a) −sec x( ) ; b) sec2 θ( ) ; c) 1; d) csc4 x( ) ; e) sin t( ); f) sin α( )
3) a) 4cos θ( ) ; b) 6sec θ( ); c) 3tan θ( )
SECTION 5.2: VERIFYING TRIGONOMETRIC IDENTITIES
1) Solutions will vary.
(Answers for Chapter 5: Analytic Trigonometry) A.5.2
SECTION 5.3: SOLVING TRIGONOMETRIC EQUATIONS
1)
a)
x ∈ x =π3+ 2πn or x =
2π3
+ 2πn n ∈( )⎧⎨⎩⎪
⎫⎬⎭⎪
. In 0, 2π[ ) : π3, 2π3
⎧⎨⎩
⎫⎬⎭
.
b) θ ∈ θ = ±
3π4
+ 2πn n ∈( )⎧⎨⎩⎪
⎫⎬⎭⎪
, or, equivalently,
θ ∈ θ =
3π4
+ 2πn or θ =5π4
+ 2πn n ∈( )⎧⎨⎩⎪
⎫⎬⎭⎪
. In 0, 2π[ ) : 3π4, 5π4
⎧⎨⎩
⎫⎬⎭
.
c) No real solutions; the solution set is ∅ . No real solutions in 0, 2π[ ) .
d)
u ∈ u =3π2
+ 2πn n ∈( )⎧⎨⎩⎪
⎫⎬⎭⎪
. Solutions in 0, 2π[ ) : 3π2
⎧⎨⎩
⎫⎬⎭
.
e)
u ∈ u =π2+ πn n ∈( )⎧
⎨⎩⎪
⎫⎬⎭⎪
. Solutions in 0, 2π[ ) : π2, 3π2
⎧⎨⎩
⎫⎬⎭
.
f)
u ∈ u =7π6
+ 2πn or u =11π
6+ 2πn n ∈( )⎧
⎨⎩⎪
⎫⎬⎭⎪
, or, equivalently,
u ∈ u =
7π6
+ 2πn or u = −π6+ 2πn n ∈( )⎧
⎨⎩⎪
⎫⎬⎭⎪
. In 0, 2π[ ) :
7π6
, 11π6
⎧⎨⎩
⎫⎬⎭
.
g)
x ∈ x = ±π3+ 2πn n ∈( )⎧
⎨⎩⎪
⎫⎬⎭⎪
, or, equivalently,
x ∈ x =
π3+ 2πn or x =
5π3
+ 2πn n ∈( )⎧⎨⎩⎪
⎫⎬⎭⎪
. In 0, 2π[ ) :
π3
, 5π3
⎧⎨⎩
⎫⎬⎭
.
h) No real solutions; the solution set is ∅ . No real solutions in 0, 2π[ ) .
i)
x ∈ x =π6+ πn n ∈( )⎧
⎨⎩⎪
⎫⎬⎭⎪
. Solutions in 0, 2π[ ) :
π6
, 7π6
⎧⎨⎩
⎫⎬⎭
.
j) θ ∈ θ =
π2+ πn n ∈( )⎧
⎨⎩⎪
⎫⎬⎭⎪
. Solutions in 0, 2π[ ) :
π2, 3π2
⎧⎨⎩
⎫⎬⎭
.
k) θ ∈ θ = ±
π6+ πn n ∈( )⎧
⎨⎩⎪
⎫⎬⎭⎪
, or, equivalently,
θ ∈ θ = π
6+πn or θ = 5π
6+πn n∈( )⎧
⎨⎩
⎫⎬⎭
. In 0,2π[ ) :
π6, 5π6, 7π6,11π6
⎧⎨⎩
⎫⎬⎭
.
(Answers for Chapter 5: Analytic Trigonometry) A.5.3