Precalculus with Limits: A Graphing Approach
Precalculus with Limits: A Graphing Approach
7th Edition
ISBN: 9781305071711
Author: Ron Larson
Publisher: Brooks/Cole
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Chapter 4.4, Problem 113E

(a)

To determine

To calculate: The exact value of function (f+g)(θ) for θ=150° , where f(θ)=sinθ and g(θ)=cosθ .

(a)

Expert Solution
Check Mark

Answer to Problem 113E

The value (f+g)(150°) is 1+32 .

Explanation of Solution

Given:

The functions f(θ)=sinθ and g(θ)=cosθ .

Concept Used:

Use the property (f+g)(θ)=f(θ)+g(θ) , for the given value of θ .

Calculation:

Substitute the values of f(θ)=sinθ and g(θ)=cosθ in above identity

  (f+g)(θ)=f(θ)+g(θ)=sinθ+cosθ , substitute θ=150° in the above equation it gives

  (f+g)(150°)=sin(150°)+cos(150°) .

By the trigonometric identity sin(θ)=sinθ and cos(θ)=cosθ , therefore sin(150°)=sin150° and cos(150°)=cos150° .

Therefore, (f+g)(150°)=sin(150°)+cos(150°)

By the trigonometric property sin(180°θ)=sinθ and cos(180°θ)=cosθ ,

Substitute θ=30° in both the properties sin(180°30°)=sin30° , it gives sin(150°)=sin30° and also cos(180°30°)=cos30° , it gives cos(150°)=cos30° .

Now substitute 12 for sin30° and 32 for cos30° in the above equation it gives,

  sin(150°)=12 and cos(150°)=32 .

Now substitute the values in the formula (f+g)(150°)=sin(150°)+cos(150°)=12+32=1+32

Therefore the value (f+g)(150°) is 1+32 .

(b)

To determine

To calculate: The exact value of function (gf)(θ) for θ=150° , where f(θ)=sinθ and g(θ)=cosθ .

(b)

Expert Solution
Check Mark

Answer to Problem 113E

The value (gf)(150°) is 132 .

Explanation of Solution

Given:

The functions f(θ)=sinθ and g(θ)=cosθ .

Concept Used:

Use the property (gf)(θ)=g(θ)f(θ) , for the given value of θ .

Calculation:

Substitute the values of f(θ)=sinθ and g(θ)=cosθ in above identity

  (gf)(θ)=g(θ)f(θ)=cosθsinθ , substitute θ=150° in the above equation it gives

  (gf)(150°)=cos(150°)sin(150°) .

By the trigonometric identity sin(θ)=sinθ and cos(θ)=cosθ , therefore sin(150°)=sin150° and cos(150°)=cos150° .

Therefore, (gf)(150°)=cos(150°)+sin(150°) .

By the trigonometric property sin(180°θ)=sinθ and cos(180°θ)=cosθ ,

Substitute θ=30° in both the properties sin(180°30°)=sin30° , it gives sin(150°)=sin30° and also cos(180°30°)=cos30° , it gives cos(150°)=cos30° .

Now substitute 12 for sin30° and 32 for cos30° in the above equation it gives,

  sin(150°)=12 and cos(150°)=32 .

Now substitute the values in the formula (gf)(150°)=cos(150°)sin(150°)=32+12=132

Therefore the value (gf)(150°) is 132 .

(c)

To determine

To calculate: The exact value of function [g(θ)]2 for θ=150° , where g(θ)=cosθ .

(c)

Expert Solution
Check Mark

Answer to Problem 113E

The value [g(150°)]2 is 34 .

Explanation of Solution

Given:

The functions f(θ)=sinθ and g(θ)=cosθ .

Concept Used:

Use the value of g(θ)=cosθ in the required value [g(θ)]2 , for the given value of θ .

Calculation:

Substitute the values of g(θ)=cosθ in above identity

  [g(θ)]2=(cosθ)2 , substitute θ=150° in the above equation it gives

  [g(150°)]2=(cos(150°))2 .

By the trigonometric identity cos(θ)=cosθ , therefore cos(150°)=cos150° .

Therefore, [g(150°)]2=(cos(150°))2 .

By the trigonometric property cos(180°θ)=cosθ ,

Substitute θ=30° in both the property cos(180°30°)=cos30° , it gives cos(150°)=cos30° .

Now substitute 32 for cos30° in the above equation it gives cos(150°)=32 .

  [g(150°)]2=(cos150°)2=(32)2=34 .

Therefore the value [g(150°)]2 is 34 .

(d)

To determine

To calculate: The exact value of function (fg)(θ) for θ=150° , where f(θ)=sinθ and g(θ)=cosθ .

(d)

Expert Solution
Check Mark

Answer to Problem 113E

The value (fg)(150°) is 34 .

Explanation of Solution

Given:

The functions f(θ)=sinθ and g(θ)=cosθ .

Concept Used:

Use the property (fg)(θ)=f(θ)g(θ) , for the given value of θ .

Calculation:

Substitute the values of f(θ)=sinθ and g(θ)=cosθ in above identity

  (fg)(θ)=f(θ)g(θ)=sinθcosθ , substitute θ=150° in the above equation it gives

  (fg)(150°)=sin(150°)cos(150°) .

By the trigonometric identity sin(θ)=sinθ and cos(θ)=cosθ , therefore sin(150°)=sin150° and cos(150°)=cos150° .

Therefore, (fg)(150°)=sin(150°)cos(150°) .

By the trigonometric property sin(180°θ)=sinθ and cos(180°θ)=cosθ ,

Substitute θ=30° in both the properties sin(180°30°)=sin30° , it gives sin(150°)=sin30° and also cos(180°30°)=cos30° , it gives cos(150°)=cos30° .

Now substitute 12 for sin30° and 32 for cos30° in the above equation it gives,

  sin(150°)=12 and cos(150°)=32 .

Now substitute the above values in the formula

  (fg)(150°)=sin(150°)cos(150°)=12×32=34 .

Therefore the value (fg)(150°) for f(θ)=sinθ and g(θ)=cosθ is 34 .

(e)

To determine

To calculate: The exact value of function f(2θ) for θ=150° , where f(θ)=sinθ .

(e)

Expert Solution
Check Mark

Answer to Problem 113E

The value f(2θ) for θ=150° is 32 .

Explanation of Solution

Given:

The functions f(θ)=sinθ and g(θ)=cosθ .

Concept Used:

Firstly calculate the angle 2θ for given θ and then calculate f(2θ) .

Calculation:

As θ=150° , so 2θ=300° .

Substitute the value of 2θ in f(2θ)=sin(2θ)

  f(300°)=sin(300°) ,

By the trigonometric property sin(θ)=sinθ , therefore sin(300°)=sin300° .

Therefore, f(300°)=sin(300°)

As sin(360°θ)=sinθ , substitute θ=60° in the equation it gives sin(360°60°)=sin60° , it gives sin300°=sin60° .

Now substitute 32 for sin60° it gives sin(300°)=32

Therefore value f(2θ) for θ=150° is 32 .

(f)

To determine

To calculate: The exact value of function g(θ) for θ=150° , where g(θ)=cosθ .

(f)

Expert Solution
Check Mark

Answer to Problem 113E

The value g(150°) is 32 .

Explanation of Solution

Given:

The functions f(θ)=sinθ and g(θ)=cosθ .

Concept Used:

Use the property g(θ)=g(θ) for the given even function g(θ)=cosθ , for the given value of θ .

Calculation:

Substitute θ=150° in the equation it gives g(θ)=g(150°)=cos(150°) .

As per trigonometric identity cos(180°θ)=cosθ , where θ is any acute angle.

Now substitute θ=30° in the above equation it gives cos(180°30°)=cos30° , which gives cos(150°)=cos30°

Substitute 32 for cos(30°) it gives

  cos(150°)=32 .

Therefore the value of cos(θ) for θ=150° is 32 .

Chapter 4 Solutions

Precalculus with Limits: A Graphing Approach

Ch. 4.1 - Prob. 11ECh. 4.1 - Prob. 12ECh. 4.1 - Prob. 13ECh. 4.1 - Prob. 14ECh. 4.1 - Prob. 15ECh. 4.1 - Prob. 16ECh. 4.1 - Prob. 17ECh. 4.1 - Prob. 18ECh. 4.1 - Prob. 19ECh. 4.1 - Prob. 20ECh. 4.1 - Prob. 21ECh. 4.1 - Prob. 22ECh. 4.1 - Prob. 23ECh. 4.1 - Prob. 24ECh. 4.1 - Prob. 25ECh. 4.1 - Prob. 26ECh. 4.1 - Prob. 27ECh. 4.1 - Prob. 28ECh. 4.1 - Prob. 29ECh. 4.1 - Prob. 30ECh. 4.1 - Prob. 31ECh. 4.1 - Prob. 32ECh. 4.1 - Prob. 33ECh. 4.1 - Prob. 34ECh. 4.1 - Prob. 35ECh. 4.1 - Prob. 36ECh. 4.1 - Prob. 37ECh. 4.1 - Prob. 38ECh. 4.1 - Prob. 39ECh. 4.1 - Prob. 40ECh. 4.1 - Prob. 41ECh. 4.1 - Prob. 42ECh. 4.1 - Prob. 43ECh. 4.1 - Prob. 44ECh. 4.1 - Prob. 45ECh. 4.1 - Prob. 46ECh. 4.1 - Prob. 47ECh. 4.1 - Prob. 48ECh. 4.1 - Prob. 49ECh. 4.1 - Prob. 50ECh. 4.1 - Prob. 51ECh. 4.1 - Prob. 52ECh. 4.1 - Prob. 53ECh. 4.1 - Prob. 54ECh. 4.1 - Prob. 55ECh. 4.1 - Prob. 56ECh. 4.1 - Prob. 57ECh. 4.1 - Prob. 58ECh. 4.1 - Prob. 59ECh. 4.1 - Prob. 60ECh. 4.1 - Prob. 61ECh. 4.1 - Prob. 62ECh. 4.1 - Prob. 63ECh. 4.1 - 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Prob. 110ECh. 4.4 - Prob. 111ECh. 4.4 - Prob. 112ECh. 4.4 - Prob. 113ECh. 4.4 - Prob. 114ECh. 4.4 - Prob. 115ECh. 4.4 - Prob. 116ECh. 4.4 - Prob. 117ECh. 4.4 - Prob. 118ECh. 4.4 - Prob. 119ECh. 4.4 - Prob. 120ECh. 4.4 - Prob. 121ECh. 4.4 - Prob. 122ECh. 4.4 - Prob. 123ECh. 4.4 - Prob. 124ECh. 4.4 - Prob. 125ECh. 4.4 - Prob. 126ECh. 4.4 - Prob. 127ECh. 4.4 - Prob. 128ECh. 4.4 - Prob. 129ECh. 4.4 - Prob. 130ECh. 4.4 - Prob. 131ECh. 4.4 - Prob. 132ECh. 4.4 - Prob. 133ECh. 4.4 - Prob. 134ECh. 4.4 - Prob. 135ECh. 4.4 - Prob. 136ECh. 4.4 - Prob. 137ECh. 4.4 - Prob. 138ECh. 4.4 - Prob. 139ECh. 4.4 - Prob. 140ECh. 4.5 - Prob. 1ECh. 4.5 - Prob. 2ECh. 4.5 - Prob. 3ECh. 4.5 - Prob. 4ECh. 4.5 - Prob. 5ECh. 4.5 - Prob. 6ECh. 4.5 - Prob. 7ECh. 4.5 - Prob. 8ECh. 4.5 - Prob. 9ECh. 4.5 - Prob. 10ECh. 4.5 - Prob. 11ECh. 4.5 - Prob. 12ECh. 4.5 - Prob. 13ECh. 4.5 - Prob. 14ECh. 4.5 - Prob. 15ECh. 4.5 - Prob. 16ECh. 4.5 - Prob. 17ECh. 4.5 - Prob. 18ECh. 4.5 - Prob. 19ECh. 4.5 - Prob. 20ECh. 4.5 - Prob. 21ECh. 4.5 - Prob. 22ECh. 4.5 - Prob. 23ECh. 4.5 - Prob. 24ECh. 4.5 - Prob. 25ECh. 4.5 - Prob. 26ECh. 4.5 - Prob. 27ECh. 4.5 - Prob. 28ECh. 4.5 - Prob. 29ECh. 4.5 - Prob. 30ECh. 4.5 - Prob. 31ECh. 4.5 - Prob. 32ECh. 4.5 - Prob. 33ECh. 4.5 - Prob. 34ECh. 4.5 - Prob. 35ECh. 4.5 - Prob. 36ECh. 4.5 - Prob. 37ECh. 4.5 - Prob. 38ECh. 4.5 - Prob. 39ECh. 4.5 - Prob. 40ECh. 4.5 - Prob. 41ECh. 4.5 - Prob. 42ECh. 4.5 - Prob. 43ECh. 4.5 - Prob. 44ECh. 4.5 - Prob. 45ECh. 4.5 - Prob. 46ECh. 4.5 - Prob. 47ECh. 4.5 - Prob. 48ECh. 4.5 - Prob. 49ECh. 4.5 - Prob. 50ECh. 4.5 - Prob. 51ECh. 4.5 - Prob. 52ECh. 4.5 - Prob. 53ECh. 4.5 - Prob. 54ECh. 4.5 - Prob. 55ECh. 4.5 - Prob. 56ECh. 4.5 - Prob. 57ECh. 4.5 - Prob. 58ECh. 4.5 - Prob. 59ECh. 4.5 - Prob. 60ECh. 4.5 - Prob. 61ECh. 4.5 - Prob. 62ECh. 4.5 - Prob. 63ECh. 4.5 - Prob. 64ECh. 4.5 - Prob. 65ECh. 4.5 - Prob. 66ECh. 4.5 - Prob. 67ECh. 4.5 - Prob. 68ECh. 4.5 - Prob. 69ECh. 4.5 - Prob. 70ECh. 4.5 - Prob. 71ECh. 4.5 - Prob. 72ECh. 4.5 - Prob. 73ECh. 4.5 - Prob. 74ECh. 4.5 - Prob. 75ECh. 4.5 - Prob. 76ECh. 4.5 - Prob. 77ECh. 4.5 - Prob. 78ECh. 4.5 - Prob. 79ECh. 4.5 - Prob. 80ECh. 4.5 - Prob. 81ECh. 4.5 - Prob. 82ECh. 4.5 - Prob. 83ECh. 4.5 - Prob. 84ECh. 4.5 - Prob. 85ECh. 4.5 - Prob. 86ECh. 4.5 - Prob. 87ECh. 4.5 - Prob. 88ECh. 4.5 - Prob. 89ECh. 4.5 - Prob. 90ECh. 4.5 - Prob. 91ECh. 4.5 - Prob. 92ECh. 4.5 - Prob. 93ECh. 4.5 - Prob. 94ECh. 4.5 - Prob. 95ECh. 4.5 - Prob. 96ECh. 4.5 - Prob. 97ECh. 4.5 - Prob. 98ECh. 4.5 - Prob. 99ECh. 4.5 - Prob. 100ECh. 4.5 - Prob. 101ECh. 4.5 - Prob. 102ECh. 4.5 - Prob. 103ECh. 4.5 - Prob. 104ECh. 4.5 - Prob. 105ECh. 4.5 - Prob. 106ECh. 4.6 - Prob. 1ECh. 4.6 - Prob. 2ECh. 4.6 - Prob. 3ECh. 4.6 - Prob. 4ECh. 4.6 - Prob. 5ECh. 4.6 - Prob. 6ECh. 4.6 - Prob. 7ECh. 4.6 - Prob. 8ECh. 4.6 - Prob. 9ECh. 4.6 - Prob. 10ECh. 4.6 - Prob. 11ECh. 4.6 - Prob. 12ECh. 4.6 - Prob. 13ECh. 4.6 - Prob. 14ECh. 4.6 - Prob. 15ECh. 4.6 - Prob. 16ECh. 4.6 - Prob. 17ECh. 4.6 - Prob. 18ECh. 4.6 - Prob. 19ECh. 4.6 - Prob. 20ECh. 4.6 - Prob. 21ECh. 4.6 - Prob. 22ECh. 4.6 - Prob. 23ECh. 4.6 - Prob. 24ECh. 4.6 - Prob. 25ECh. 4.6 - Prob. 26ECh. 4.6 - Prob. 27ECh. 4.6 - Prob. 28ECh. 4.6 - Prob. 29ECh. 4.6 - Prob. 30ECh. 4.6 - Prob. 31ECh. 4.6 - Prob. 32ECh. 4.6 - Prob. 33ECh. 4.6 - Prob. 34ECh. 4.6 - Prob. 35ECh. 4.6 - Prob. 36ECh. 4.6 - Prob. 37ECh. 4.6 - Prob. 38ECh. 4.6 - Prob. 39ECh. 4.6 - Prob. 40ECh. 4.6 - Prob. 41ECh. 4.6 - Prob. 42ECh. 4.6 - Prob. 43ECh. 4.6 - Prob. 44ECh. 4.6 - Prob. 45ECh. 4.6 - Prob. 46ECh. 4.6 - Prob. 47ECh. 4.6 - Prob. 48ECh. 4.6 - Prob. 49ECh. 4.6 - Prob. 50ECh. 4.6 - Prob. 51ECh. 4.6 - Prob. 52ECh. 4.6 - Prob. 53ECh. 4.6 - Prob. 54ECh. 4.6 - Prob. 55ECh. 4.6 - Prob. 56ECh. 4.6 - Prob. 57ECh. 4.6 - Prob. 58ECh. 4.6 - Prob. 59ECh. 4.6 - Prob. 60ECh. 4.6 - Prob. 61ECh. 4.6 - Prob. 62ECh. 4.6 - Prob. 63ECh. 4.6 - Prob. 64ECh. 4.6 - Prob. 65ECh. 4.6 - Prob. 66ECh. 4.6 - Prob. 68ECh. 4.6 - Prob. 69ECh. 4.6 - Prob. 70ECh. 4.6 - Prob. 71ECh. 4.6 - Prob. 72ECh. 4.6 - Prob. 73ECh. 4.6 - Prob. 74ECh. 4.6 - Prob. 75ECh. 4.6 - Prob. 76ECh. 4.6 - Prob. 77ECh. 4.6 - Prob. 78ECh. 4.6 - Prob. 79ECh. 4.6 - Prob. 80ECh. 4.6 - Prob. 81ECh. 4.6 - Prob. 82ECh. 4.6 - Prob. 83ECh. 4.6 - Prob. 84ECh. 4.6 - Prob. 85ECh. 4.6 - Prob. 86ECh. 4.6 - Prob. 87ECh. 4.6 - Prob. 88ECh. 4.6 - Prob. 89ECh. 4.6 - Prob. 90ECh. 4.7 - Prob. 1ECh. 4.7 - Prob. 2ECh. 4.7 - Prob. 3ECh. 4.7 - Prob. 4ECh. 4.7 - Prob. 5ECh. 4.7 - Prob. 6ECh. 4.7 - Prob. 7ECh. 4.7 - Prob. 8ECh. 4.7 - Prob. 9ECh. 4.7 - Prob. 10ECh. 4.7 - Prob. 11ECh. 4.7 - Prob. 12ECh. 4.7 - Prob. 13ECh. 4.7 - Prob. 14ECh. 4.7 - Prob. 15ECh. 4.7 - Prob. 16ECh. 4.7 - Prob. 17ECh. 4.7 - Prob. 18ECh. 4.7 - Prob. 19ECh. 4.7 - Prob. 20ECh. 4.7 - Prob. 21ECh. 4.7 - Prob. 22ECh. 4.7 - Prob. 23ECh. 4.7 - Prob. 24ECh. 4.7 - Prob. 25ECh. 4.7 - Prob. 26ECh. 4.7 - Prob. 27ECh. 4.7 - Prob. 28ECh. 4.7 - Prob. 29ECh. 4.7 - Prob. 30ECh. 4.7 - Prob. 31ECh. 4.7 - Prob. 32ECh. 4.7 - Prob. 33ECh. 4.7 - Prob. 34ECh. 4.7 - Prob. 35ECh. 4.7 - Prob. 36ECh. 4.7 - Prob. 37ECh. 4.7 - Prob. 38ECh. 4.7 - Prob. 39ECh. 4.7 - Prob. 40ECh. 4.7 - Prob. 41ECh. 4.7 - Prob. 42ECh. 4.7 - Prob. 43ECh. 4.7 - Prob. 44ECh. 4.7 - Prob. 45ECh. 4.7 - Prob. 46ECh. 4.7 - Prob. 47ECh. 4.7 - Prob. 48ECh. 4.7 - Prob. 49ECh. 4.7 - Prob. 50ECh. 4.7 - Prob. 51ECh. 4.7 - Prob. 52ECh. 4.7 - Prob. 53ECh. 4.7 - Prob. 54ECh. 4.7 - Prob. 55ECh. 4.7 - Prob. 56ECh. 4.7 - Prob. 57ECh. 4.7 - Prob. 58ECh. 4.7 - Prob. 59ECh. 4.7 - Prob. 60ECh. 4.7 - Prob. 61ECh. 4.7 - Prob. 62ECh. 4.7 - Prob. 63ECh. 4.7 - Prob. 64ECh. 4.7 - Prob. 65ECh. 4.7 - Prob. 66ECh. 4.7 - Prob. 67ECh. 4.7 - Prob. 68ECh. 4.7 - Prob. 69ECh. 4.7 - Prob. 70ECh. 4.7 - Prob. 71ECh. 4.7 - Prob. 72ECh. 4.7 - Prob. 73ECh. 4.7 - Prob. 74ECh. 4.7 - Prob. 75ECh. 4.7 - Prob. 76ECh. 4.7 - Prob. 77ECh. 4.7 - Prob. 78ECh. 4.7 - Prob. 79ECh. 4.7 - Prob. 80ECh. 4.7 - Prob. 81ECh. 4.7 - Prob. 82ECh. 4.7 - Prob. 83ECh. 4.7 - Prob. 84ECh. 4.7 - Prob. 85ECh. 4.7 - Prob. 86ECh. 4.7 - Prob. 87ECh. 4.7 - Prob. 88ECh. 4.7 - Prob. 89ECh. 4.7 - Prob. 90ECh. 4.7 - Prob. 91ECh. 4.7 - Prob. 92ECh. 4.7 - Prob. 93ECh. 4.7 - Prob. 94ECh. 4.7 - Prob. 95ECh. 4.7 - Prob. 96ECh. 4.7 - Prob. 97ECh. 4.7 - Prob. 98ECh. 4.7 - Prob. 99ECh. 4.7 - Prob. 100ECh. 4.7 - Prob. 101ECh. 4.7 - Prob. 102ECh. 4.7 - Prob. 103ECh. 4.7 - Prob. 104ECh. 4.7 - Prob. 105ECh. 4.7 - Prob. 106ECh. 4.7 - Prob. 107ECh. 4.7 - Prob. 108ECh. 4.7 - Prob. 109ECh. 4.7 - Prob. 110ECh. 4.7 - Prob. 111ECh. 4.7 - Prob. 112ECh. 4.7 - Prob. 113ECh. 4.7 - Prob. 114ECh. 4.7 - Prob. 115ECh. 4.7 - Prob. 116ECh. 4.7 - Prob. 117ECh. 4.7 - Prob. 118ECh. 4.7 - Prob. 119ECh. 4.7 - Prob. 120ECh. 4.7 - Prob. 121ECh. 4.7 - Prob. 122ECh. 4.7 - Prob. 123ECh. 4.7 - Prob. 124ECh. 4.7 - Prob. 125ECh. 4.7 - Prob. 126ECh. 4.8 - Prob. 1ECh. 4.8 - Prob. 2ECh. 4.8 - Prob. 3ECh. 4.8 - Prob. 4ECh. 4.8 - Prob. 5ECh. 4.8 - Prob. 6ECh. 4.8 - Prob. 7ECh. 4.8 - Prob. 8ECh. 4.8 - Prob. 9ECh. 4.8 - Prob. 10ECh. 4.8 - Prob. 11ECh. 4.8 - Prob. 12ECh. 4.8 - Prob. 13ECh. 4.8 - Prob. 14ECh. 4.8 - Prob. 15ECh. 4.8 - Prob. 16ECh. 4.8 - Prob. 17ECh. 4.8 - Prob. 18ECh. 4.8 - Prob. 19ECh. 4.8 - Prob. 20ECh. 4.8 - Prob. 21ECh. 4.8 - Prob. 22ECh. 4.8 - Prob. 23ECh. 4.8 - Prob. 24ECh. 4.8 - Prob. 25ECh. 4.8 - Prob. 26ECh. 4.8 - Prob. 27ECh. 4.8 - Prob. 28ECh. 4.8 - Prob. 29ECh. 4.8 - Prob. 30ECh. 4.8 - Prob. 31ECh. 4.8 - Prob. 32ECh. 4.8 - Prob. 33ECh. 4.8 - Prob. 34ECh. 4.8 - Prob. 35ECh. 4.8 - Prob. 36ECh. 4.8 - Prob. 37ECh. 4.8 - Prob. 38ECh. 4.8 - Prob. 39ECh. 4.8 - Prob. 40ECh. 4.8 - Prob. 41ECh. 4.8 - Prob. 42ECh. 4.8 - Prob. 43ECh. 4.8 - Prob. 44ECh. 4.8 - Prob. 45ECh. 4.8 - Prob. 46ECh. 4.8 - Prob. 47ECh. 4.8 - Prob. 48ECh. 4.8 - Prob. 49ECh. 4.8 - Prob. 50ECh. 4.8 - Prob. 51ECh. 4.8 - Prob. 52ECh. 4.8 - Prob. 53ECh. 4.8 - Prob. 54ECh. 4.8 - Prob. 55ECh. 4.8 - Prob. 56ECh. 4.8 - Prob. 57ECh. 4.8 - Prob. 58ECh. 4.8 - Prob. 59ECh. 4.8 - Prob. 60ECh. 4.8 - Prob. 61ECh. 4.8 - Prob. 62ECh. 4.8 - Prob. 63ECh. 4.8 - Prob. 64ECh. 4.8 - Prob. 65ECh. 4.8 - Prob. 66ECh. 4.8 - Prob. 67ECh. 4.8 - Prob. 68ECh. 4.8 - Prob. 69ECh. 4.8 - Prob. 70ECh. 4.8 - Prob. 71ECh. 4.8 - Prob. 72ECh. 4.8 - Prob. 73ECh. 4.8 - Prob. 74ECh. 4.8 - Prob. 75ECh. 4.8 - Prob. 76ECh. 4.8 - Prob. 77ECh. 4.8 - Prob. 78ECh. 4.8 - Prob. 79ECh. 4.8 - Prob. 80ECh. 4.8 - Prob. 81ECh. 4.8 - Prob. 82ECh. 4.8 - Prob. 83ECh. 4.8 - Prob. 84ECh. 4 - Prob. 1RECh. 4 - Prob. 2RECh. 4 - Prob. 3RECh. 4 - Prob. 4RECh. 4 - Prob. 5RECh. 4 - Prob. 6RECh. 4 - Prob. 7RECh. 4 - Prob. 8RECh. 4 - Prob. 9RECh. 4 - Prob. 10RECh. 4 - Prob. 11RECh. 4 - Prob. 12RECh. 4 - Prob. 13RECh. 4 - Prob. 14RECh. 4 - Prob. 15RECh. 4 - Prob. 16RECh. 4 - Prob. 17RECh. 4 - Prob. 18RECh. 4 - Prob. 19RECh. 4 - Prob. 20RECh. 4 - Prob. 21RECh. 4 - Prob. 22RECh. 4 - Prob. 23RECh. 4 - Prob. 24RECh. 4 - Prob. 25RECh. 4 - Prob. 26RECh. 4 - Prob. 27RECh. 4 - Prob. 28RECh. 4 - Prob. 29RECh. 4 - Prob. 30RECh. 4 - Prob. 31RECh. 4 - Prob. 32RECh. 4 - Prob. 33RECh. 4 - Prob. 34RECh. 4 - Prob. 35RECh. 4 - Prob. 36RECh. 4 - Prob. 37RECh. 4 - Prob. 38RECh. 4 - Prob. 39RECh. 4 - Prob. 40RECh. 4 - Prob. 41RECh. 4 - Prob. 42RECh. 4 - Prob. 43RECh. 4 - Prob. 44RECh. 4 - Prob. 45RECh. 4 - Prob. 46RECh. 4 - Prob. 47RECh. 4 - Prob. 48RECh. 4 - Prob. 49RECh. 4 - Prob. 50RECh. 4 - Prob. 51RECh. 4 - Prob. 52RECh. 4 - Prob. 53RECh. 4 - Prob. 54RECh. 4 - Prob. 55RECh. 4 - Prob. 56RECh. 4 - Prob. 57RECh. 4 - Prob. 58RECh. 4 - Prob. 59RECh. 4 - Prob. 60RECh. 4 - Prob. 61RECh. 4 - Prob. 62RECh. 4 - Prob. 63RECh. 4 - Prob. 64RECh. 4 - Prob. 65RECh. 4 - Prob. 66RECh. 4 - Prob. 67RECh. 4 - Prob. 68RECh. 4 - Prob. 69RECh. 4 - Prob. 70RECh. 4 - Prob. 71RECh. 4 - Prob. 72RECh. 4 - Prob. 73RECh. 4 - Prob. 74RECh. 4 - Prob. 75RECh. 4 - Prob. 76RECh. 4 - Prob. 77RECh. 4 - Prob. 78RECh. 4 - Prob. 79RECh. 4 - Prob. 80RECh. 4 - Prob. 81RECh. 4 - Prob. 82RECh. 4 - Prob. 83RECh. 4 - Prob. 84RECh. 4 - Prob. 85RECh. 4 - Prob. 86RECh. 4 - Prob. 87RECh. 4 - Prob. 88RECh. 4 - Prob. 89RECh. 4 - Prob. 90RECh. 4 - Prob. 91RECh. 4 - Prob. 92RECh. 4 - Prob. 93RECh. 4 - Prob. 94RECh. 4 - Prob. 95RECh. 4 - Prob. 96RECh. 4 - Prob. 97RECh. 4 - Prob. 98RECh. 4 - Prob. 99RECh. 4 - Prob. 100RECh. 4 - Prob. 101RECh. 4 - Prob. 102RECh. 4 - Prob. 103RECh. 4 - Prob. 104RECh. 4 - Prob. 105RECh. 4 - Prob. 106RECh. 4 - Prob. 107RECh. 4 - Prob. 108RECh. 4 - Prob. 109RECh. 4 - Prob. 110RECh. 4 - Prob. 111RECh. 4 - Prob. 112RECh. 4 - Prob. 113RECh. 4 - Prob. 114RECh. 4 - Prob. 115RECh. 4 - Prob. 116RECh. 4 - Prob. 117RECh. 4 - Prob. 118RECh. 4 - Prob. 119RECh. 4 - Prob. 120RECh. 4 - Prob. 121RECh. 4 - Prob. 122RECh. 4 - Prob. 123RECh. 4 - Prob. 124RECh. 4 - Prob. 125RECh. 4 - Prob. 126RECh. 4 - Prob. 127RECh. 4 - Prob. 128RECh. 4 - Prob. 129RECh. 4 - Prob. 130RECh. 4 - Prob. 131RECh. 4 - Prob. 132RECh. 4 - Prob. 133RECh. 4 - Prob. 134RECh. 4 - Prob. 135RECh. 4 - Prob. 136RECh. 4 - Prob. 137RECh. 4 - Prob. 138RECh. 4 - Prob. 139RECh. 4 - Prob. 140RECh. 4 - Prob. 141RECh. 4 - Prob. 142RECh. 4 - Prob. 143RECh. 4 - Prob. 144RECh. 4 - Prob. 145RECh. 4 - Prob. 146RECh. 4 - Prob. 147RECh. 4 - Prob. 148RECh. 4 - Prob. 149RECh. 4 - Prob. 150RECh. 4 - Prob. 151RECh. 4 - Prob. 152RECh. 4 - Prob. 153RECh. 4 - Prob. 154RECh. 4 - Prob. 155RECh. 4 - Prob. 156RECh. 4 - Prob. 157RECh. 4 - Prob. 158RECh. 4 - Prob. 159RECh. 4 - Prob. 160RECh. 4 - Prob. 161RECh. 4 - Prob. 162RECh. 4 - Prob. 163RECh. 4 - Prob. 164RECh. 4 - Prob. 165RECh. 4 - Prob. 166RECh. 4 - Prob. 167RECh. 4 - Prob. 168RECh. 4 - Prob. 169RECh. 4 - Prob. 170RECh. 4 - Prob. 171RECh. 4 - Prob. 172RECh. 4 - Prob. 173RECh. 4 - Prob. 174RECh. 4 - Prob. 175RECh. 4 - Prob. 176RECh. 4 - Prob. 177RECh. 4 - Prob. 178RECh. 4 - Prob. 179RECh. 4 - Prob. 180RECh. 4 - Prob. 181RECh. 4 - Prob. 182RECh. 4 - Prob. 183RECh. 4 - Prob. 184RECh. 4 - Prob. 185RECh. 4 - Prob. 186RECh. 4 - Prob. 187RECh. 4 - Prob. 188RECh. 4 - Prob. 189RECh. 4 - Prob. 190RECh. 4 - Prob. 191RECh. 4 - Prob. 192RECh. 4 - Prob. 193RECh. 4 - Prob. 194RECh. 4 - Prob. 196RECh. 4 - Prob. 197RECh. 4 - Prob. 1CTCh. 4 - Prob. 2CTCh. 4 - Prob. 3CTCh. 4 - Prob. 4CTCh. 4 - Prob. 5CTCh. 4 - Prob. 6CTCh. 4 - Prob. 7CTCh. 4 - Prob. 8CTCh. 4 - Prob. 9CTCh. 4 - Prob. 10CTCh. 4 - Prob. 11CTCh. 4 - Prob. 12CTCh. 4 - Prob. 13CTCh. 4 - Prob. 14CTCh. 4 - Prob. 15CTCh. 4 - Prob. 16CTCh. 4 - Prob. 17CTCh. 4 - Prob. 18CTCh. 4 - Prob. 19CTCh. 4 - Prob. 20CTCh. 4 - Prob. 21CTCh. 4 - Prob. 22CTCh. 4 - Prob. 23CTCh. 4 - Prob. 24CTCh. 4 - Prob. 1PFRCh. 4 - Prob. 2PFRCh. 4 - Prob. 3PFRCh. 4 - Prob. 4PFRCh. 4 - Prob. 5PFRCh. 4 - Prob. 6PFRCh. 4 - Prob. 7PFRCh. 4 - Prob. 8PFRCh. 4 - Prob. 9PFRCh. 4 - Prob. 10PFRCh. 4 - Prob. 11PFRCh. 4 - Prob. 12PFRCh. 4 - Prob. 13PFRCh. 4 - Prob. 14PFRCh. 4 - Prob. 15PFRCh. 4 - Prob. 16PFRCh. 4 - Prob. 17PFRCh. 4 - Prob. 18PFRCh. 4 - Prob. 19PFRCh. 4 - Prob. 20PFRCh. 4 - Prob. 21PFRCh. 4 - Prob. 22PFRCh. 4 - Prob. 23PFRCh. 4 - Prob. 24PFR
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