Skip to document
Welcome to StudocuSign in to access the best study resources
Guest user
0followers
HomeMy LibraryAsk AI
  • You don't have any recent items yet.
My Library
  • You don't have any courses yet.
  • You don't have any books yet.
  • You don't have any Studylists yet.

AF12 Chapter 5 Solutions

wwe
Course

Mathematics Fundamentals (MATH 020)

999+ Documents
Students shared 3475 documents in this course
Academic year: 2022/2023

Comments

Please sign in or register to post comments.

Students also viewed

Related documents

Preview text

Chapter 5 Trigonometric Functions

Chapter 5 Prerequisite Skills

Chapter 5 Prerequisite Skills Question 1 Page 250

a) 0 b) 0.

c) –5 d) –0.

Chapter 5 Prerequisite Skills Question 2 Page 250

a) 5 b) 32.

c) 0 d) –1.

Chapter 5 Prerequisite Skills Question 3 Page 250

a) sin 5 4! = sin "! + ! 4 #$

%

&'

= ( sin ! 4

= ( 1 2

b) cos ! 6 = 23

c) tan 3 ! 4

= tan ! 2

+ !

4

"

#$

%

&'

= ( cot ! 4 = ( 1

d) sin 5 ! 6

= sin 3! 6

+ 2!

6

"

#$

%

&'

= sin ! 2

+ !

3

"

#$

%

&'

= cos ! 3 = 1 2

e) cos 5 ! 3

= cos 6! 3

" !

3

null

$%

&

'(

= cos # 2! " ! 3 $%

&

'(

= cos ! 3

= 12

f) tan 4 ! 3

= tan! + ! 3

"

#$

%

&'

= tan ! 3 = 3

Chapter 5 Prerequisite Skills Question 4 Page 250

a) csc 5 ! 3

= csc 2! " ! 3

null

$%

&

'(

= " csc ! 3 = " 2 3

b) sec 7 ! 6

= sec! + ! 6

"

#$

%

&'

= ( sec ! 6 = ( 2 3

c) cot 7 ! 4

= cot 2! " ! 4

null

$%

&

'(

= " cot ! 4 = " 1

d) sec 2! = sec 0 = 1

e) cot 3 2! = cot ! 2

= 0

f) csc ! 4 = 2

Chapter 5 Prerequisite Skills Question 5 Page 250

y = sin x

Chapter 5 Prerequisite Skills Question 6 Page 250

y = cos x

Chapter 5 Prerequisite Skills Question 7 Page 250

The graphs of the sine and cosine functions are periodic because they repeat a pattern of y-values at regular intervals of their domain.

Chapter 5 Prerequisite Skills Question 9 Page 250

a) Amplitude is 2; period is 360°; phase shift of 90° left; vertical translation of 1 unit up.

b) Maximum 2 + 1 = 3, Minimum –2 + 1 = –

c) 0 = 2cos(x + 90 °) + 1 ! 1 = 2cos(x + 90 °) ! 1 2

= cos(x + 90 °)

cos! 1! 1 2

"

#$

%

&'

= x + 90 ° 120 °! 90 ° = x x = 30 ° 180 °! x = 150 ° 360 ° + x = 390 °

The first three x-intercepts are 30°, 150°, and 390°.

d) y = 2cos( 0 + 90 °) + 1 y = 2(0) + 1 y = 1

The y-intercept is 1.

Chapter 5 Prerequisite Skills Question 10 Page 250

a) x = 31°

b) x = 141°

c) x = 74°

d) x = 27°

Chapter 5 Prerequisite Skills Question 11 Page 250

a) x = 0.

b) x = 2.

c) x = 0.

d) x = 0.

Chapter 5 Prerequisite Skills Question 12 Page 251

a) x 2 + x! 2 = 0 (x + 2)(x! 1) = 0

The equations of the vertical asymptotes are x = –2 and x = 1.

b) y = 0

c)

Chapter 5 Prerequisite Skills Question 13 Page 251

a) The instantaneous rate of change at x = 2 is 3. The function is linear so the rate of change is the slope of the line.

b) The instantaneous rate of change is the same as the average rate of change.

Chapter 5 Prerequisite Skills Question 14 Page 251

a) 6! 60 25

= 14.

Justine rode at a rate of 14 km/h.

Chapter 5 Section 1 Graphs of Sine, Cosine, and Tangent Functions

Chapter 5 Section 1 Question 1 Page 258

a) The maximum value is y = 4 + 1 = 5; the values of x where it occurs are x =! 3 " 2

and " 2

.

Maxima:! 3 " 2

# , 5

$%

&

'(

, !

2

" , 5

#$

%

&'

The minimum value is 4 + (–1) = 3; the values of x where it occurs are x =! " 2

and 3 " 2

.

Minima:! " 2

# , 3

$%

&

'(

, 3!

2

" , 3

#$

%

&'

b) The maximum value is y = –5 + 1 or –4. The values of x where it occurs are x = –2π, 0, and 2π. Maxima: (–2π, –4), (0, –4), (2π, –4)

The minimum value is y = –5 + (–1) or –6. The values of x where it occurs are x = –π and π. Minima: (–π, –6), (π, –6)

c) The maximum value is y = 1 – 2 or –1.

The values of x where it occurs are x =! 32 " and " 2.

Maxima: #! 32 " , – 1 $%

&

'(

, " ! 2 , – 1

#$

%

&'

The minimum value is y = –1 – 2 or –3. The values of x where it occurs is x =! " 2

and 3 " 2

.

Minima: – ! 2

" , – 3

#$

%

&'

, 3!

2

# ," 3

$%

&

'(

d) The maximum value is y = 1 + 1 or 2. The values of x where it occurs are x = –2π, 0, and 2π. Maxima: (–2π, 2), (0, 2), (2π, 2) The minimum value is y = –1 + 1 or 0. The values of x where it occurs are x = –π and π. Minima: (–π, 0), (π, 0)

Chapter 5 Section 1 Question 2 Page 258

a) b)

c) d)

Chapter 5 Section 1 Question 3 Page 258

a) y = 3sin x

b) y = 5cos x

c) y = !4sin x

d) y = !2cos x

Chapter 5 Section 1 Question 4 Page 258

a) b)

c) d)

b) k = 2! 3! 2

= 4

3

So, y = cos 4 3

x.

c) k = 26!! = 13

So, y = sin 1 3 x

d) k = 2! !

= 2

So, y = cos 2x

Chapter 5 Section 1 Question 8 Page 258

a) b)

c) d)

Chapter 5 Section 1 Question 9 Page 258

a) k = 2! ! = 2

So, y = 3sin 2x

b)

Chapter 5 Section 1 Question 10 Page 258

a) a = 7! 2 (! 3)= 5

The amplitude is 5.

b) 7 – 5 = 2 The vertical translation is up 2 units.

c)

Chapter 5 Section 1 Question 11 Page 258

a) Period = ! 3 " #" ! 6 $%

&

'(

= ! 2

b) Since one cycle of y = sin x begins at x = 0, this function has a phase shift of ! 6

to the left.

c) k = 2! ! 2

= 4

So, y = sin 4 x + ! 6

"

#$

%

&'

(

)

*

+

,

-

d)

b) The function is even. The graph of y = cos(!x) is equivalent to the graph of y = cos x. Proof: L. = cos(!x) = cos(0! x) = cos 0cos x + sin 0sin x = cos x

R. = cos x

L. = R.

c) The function is odd. The graph of y = tan(!x) is equivalent to the graph of y =! tan x. Proof: L. = tan(!x) = tan(0! x) = tan 0 ! tan x 1 + tan 0 tan x = 0! tan x 1 + 0 =! tan x

R. = –tan x

L. = R.

Chapter 5 Section 1 Question 15 Page 259

Answers may vary. Sample answer:

Where y = cos x and y = sin x intersect, y = tan x equals 1. Where y = cos x equals zero, y = tan x is undefined. Where y = sin x equals zero, y = tan x equals 0.

Chapter 5 Section 1 Question 16 Page 259

a)-f)

4

-1 0

f x)

xA =

A

g) For positive xA, the amplitude gets larger as xA gets larger. For negative xA, the amplitude gets larger as xA gets larger, but sin x is reflected in the x-axis.

h) The amplitude range changes if you use a circle other than a unit circle.

4

-1 0

g ( )x

xB =

f x)

xA =

A B

i)

4

-1 0

q ( )x

f x)

xA =

A

Chapter 5 Section 1 Question 18 Page 259

a) period = 60 20

= 3 s/cycle

k = 2! 3 a = 1 2 = 0.

d = 0 " 23! t #$

%

&'

b)

c) period = 60 30

= 2 s/cycle

k = 22! =! The amplitude remains the same. The waves will be closer together. The equation becomes d = 0( !t).

Chapter 5 Section 1 Question 19 Page 260

The solutions for Achievement Check can be found in the Teacher’s Resource.

Chapter 5 Section 1 Question 20 Page 260

a) Answers may vary. For example, I predict that there will be no values of the function below the x-axis.

b)

c) The relation is a function since it satisfies the vertical line test.

d) The relation is even; it is symmetrical about the y–axis. Prove that sin(!x) = sin x. L. = sin(!x) R. = sin x = sin(0! x) = sin 0cos x! sin x cos 0 =! sin x = sin x

Since L. = R., y = sin x is an even function.

Chapter 5 Section 1 Question 21 Page 261

a) b)

c) d)

ii) x = r cos! y = r sin!

3 x + 4 y = 5 3 r cos! + 4 r sin! = 5 r(3cos! + 4sin! ) = 5 r = 3cos! 5 + 4sin!

iii) x 2 + y 2! 4 y = x 2 + y 2 r 2! 4 r sin " = r 2 (x 2 + y 2 = r 2 , y = r sin " ) r 2 = r + 4 r sin " r 2 = r(1+ 4sin " ) r = 1 + 4sin "

b) i) x 2 + y 2 = 36 (r 2 = x 2 + y 2 )

ii) x = r cos! cos! = x r r = 3cos! r = 3 x r

"

#$

%

&'

r 2 = 3 x x 2 + y 2 = 3 x (r 2 = x 2 + y 2 )

iii) y = r sin! x = r cos! y r

= sin! x = r cos!

r = 2sin! + 2cos! r = 2 "yr #$

%

&'

  • 2 "xr #$

%

&'

r = 2 y +r 2 x r 2 = 2 y + 2 x (r 2 = x 2 + y 2 ) x 2 + y 2 = 2 x + 2 y

Chapter 5 Section 2 Graphs of Reciprocal Trigonometric Functions

Chapter 5 Section 2 Question 1 Page 267

The values of x such that csc x = 5 are x =& 0 and x =& 2..

Chapter 5 Section 2 Question 2 Page 267

The values of x such that sec x = 2 are x =& 1 and x =& 5..

Chapter 5 Section 2 Question 3 Page 267

The values of x such that cot x =! 4 are x =& 2 and x =& 6..

Chapter 5 Section 2 Question 4 Page 267

a) Answers may vary. For example, the cosecant function is the reciprocal of the sine function and sin! 1 is the opposite operation of sine.

b) csc 1 2

=& 1.

sin! 1 2

"

#$

%

&'

= (

4

or about 0.

Was this document helpful?

AF12 Chapter 5 Solutions

Course: Mathematics Fundamentals (MATH 020)

999+ Documents
Students shared 3475 documents in this course

University: Istituto di Istruzione Superiore Mariano IV d'Arborea

Was this document helpful?
MHR
•
Advanced Functions 12 Solutions 474
Chapter 5 Trigonometric Functions
Chapter 5 Prerequisite Skills
Chapter 5 Prerequisite Skills Question 1 Page 250
a) 0.5878 b) 0.9659
c) –5.6713 d) –0.4142
Chapter 5 Prerequisite Skills Question 2 Page 250
a) 5.9108 b) 32.4765
c) 0.3773 d) –1.4479
Chapter 5 Prerequisite Skills Question 3 Page 250
a)
sin 5
!
4
=sin
!
+
!
4
"
#
$%
&
'
=(sin
!
4
=(1
2
b)
cos
!
6
=
3
2
c)
tan 3
!
4
=tan
!
2
+
!
4
"
#
$%
&
'
=(cot
!
4
=(1
d)
sin 5
!
6
=sin 3
!
6
+2
!
6
"
#
$%
&
'
=sin
!
2
+
!
3
"
#
$%
&
'
=cos
!
3
=1
2
e)
cos 5
!
3
=cos 6
!
3"
!
3
#
$
%&
'
(
=cos 2
!
"
!
3
#
$
%&
'
(
=cos
!
3
=1
2
f)
tan 4
!
3
=tan
!
+
!
3
"
#
$%
&
'
=tan
!
3
=3

Students also viewed

Related documents