Precalculus with Limits: A Graphing Approach

7th Edition

ISBN: 9781305071711

Author: Ron Larson

Publisher: Brooks/Cole

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Question

Chapter 4.4, Problem 114E

(a)

To determine

To calculate: The exact value of function $(f+g)(θ)$ for $θ=−300°$ , where $f(θ)=sinθ$ and $g(θ)=cosθ$ .

(a)

Expert Solution

Answer to Problem 114E

The value $(f+g)(−300°)$ is $3+12$ .

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 $θ=−300°$ in the above equation it gives

$(f+g)(−300°)=sin(−300°)+cos(−300°)$ .

By the trigonometric identity $sin(−θ)=−sinθ$ and $cos(−θ)=cosθ$ , therefore $sin(−300°)=−sin300°$ and $cos(−300°)=cos300°$ .

Therefore, $(f+g)(−300°)=−sin(300°)+cos(300°)$

By the trigonometric property $sin(360°−θ)=−sinθ$ and $cos(360°−θ)=cosθ$ ,

Substitute $θ=60°$ in both the properties $sin(360°−60°)=−sin60°$ , it gives $sin(300°)=−sin60°$ and also $cos(360°−60°)=cos60°$ , it gives $cos(300°)=cos60°$ .

Now substitute $12$ for $cos60°$ and $32$ for $sin60°$ in the above equation it gives,

$sin(300°)=−32$ and $cos(300°)=12$ .

Now substitute the values in the formula $(f+g)(−300°)=sin(−300°)+cos(−300°)=32+12=3+12$

Therefore the value $(f+g)(−300°)$ is $3+12$ .

(b)

To determine

To calculate: The exact value of function $(g−f)(θ)$ for $θ=−300°$ , where $f(θ)=sinθ$ and $g(θ)=cosθ$ .

(b)

Expert Solution

Answer to Problem 114E

The value $(g−f)(−300°)$ is $1−32$ .

Explanation of Solution

Given:

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

Concept Used:

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

Calculation:

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

$(g−f)(θ)=g(θ)−f(θ)=cosθ−sinθ$ , substitute $θ=−300°$ in the above equation it gives

$(g−f)(−300°)=cos(−300°)−sin(−300°)$ .

By the trigonometric identity $sin(−θ)=−sinθ$ and $cos(−θ)=cosθ$ , therefore $sin(−300°)=−sin300°$ and $cos(−300°)=cos300°$ .

Therefore, $(g−f)(−300°)=cos(300°)+sin(300°)$

By the trigonometric property $sin(360°−θ)=−sinθ$ and $cos(360°−θ)=cosθ$ ,

Substitute $θ=60°$ in both the properties $sin(360°−60°)=−sin60°$ , it gives $sin(300°)=−sin60°$ and also $cos(360°−60°)=cos60°$ , it gives $cos(300°)=cos60°$ .

Now substitute $12$ for $cos60°$ and $32$ for $sin60°$ in the above equation it gives,

$sin(300°)=−32$ and $cos(300°)=12$ .

Now substitute the values in the formula $(g−f)(−300°)=cos(−300°)−sin(−300°)=12−32=1−32$

Therefore the value $(g−f)(−300°)$ is $1−32$ .

(c)

To determine

To calculate: The exact value of function $[g(θ)]2$ for $θ=−300°$ , where $g(θ)=cosθ$ .

(c)

Expert Solution

Answer to Problem 114E

The value $[g(−300°)]2$ is $14$ .

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 $θ=−300°$ in the above equation it gives

$[g(−300°)]2=(cos(−300°))2$ .

By the trigonometric identity $cos(−θ)=cosθ$ , therefore $cos(−300°)=cos300°$ .

Therefore, $[g(−300°)]2=(cos(300°))2$ .

By the trigonometric property $cos(360°−θ)=cosθ$ ,

Substitute $θ=60°$ in both the property $cos(360°−60°)=cos60°$ , it gives $cos(300°)=cos60°$ .

Now substitute $12$ for $cos60°$ in the above equation it gives $cos(300°)=12$ .

$[g(−300°)]2=(cos300°)2=(12)2=14$ .

Therefore the value $[g(−300°)]2$ is $14$ .

(d)

To determine

To calculate: The exact value of function $(fg)(θ)$ for $θ=−300°$ , where $f(θ)=sinθ$ and $g(θ)=cosθ$ .

(d)

Expert Solution

Answer to Problem 114E

The value $(fg)(−300°)$ 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 $θ=−300°$ in the above equation it gives

$(fg)(−300°)=sin(−300°)cos(−300°)$ .

By the trigonometric identity $sin(−θ)=−sinθ$ and $cos(−θ)=cosθ$ , therefore $sin(−300°)=−sin300°$ and $cos(−300°)=cos300°$ .

Therefore, $(fg)(−300°)=−sin(300°)cos(300°)$ .

By the trigonometric property $sin(360°−θ)=−sinθ$ and $cos(360°−θ)=cosθ$ ,

Substitute $θ=60°$ in both the properties $sin(360°−60°)=−sin60°$ , it gives $sin(300°)=−sin60°$ and also $cos(360°−60°)=cos60°$ , it gives $cos(300°)=cos60°$ .

Now substitute $12$ for $cos60°$ and $32$ for $sin60°$ in the above equation it gives,

$sin(300°)=−32$ and $cos(300°)=12$ .

Now substitute the above values in the formula

$(fg)(−300°)=−sin(300°)cos(300°)=−(−32)×12=34$ .

Therefore the value $(fg)(−300°)$ for $f(θ)=sinθ$ and $g(θ)=cosθ$ is $34$ .

(e)

To determine

To calculate: The exact value of function $f(2θ)$ for $θ=−300°$ , where $f(θ)=sinθ$ .

(e)

Expert Solution

Answer to Problem 114E

The value $f(2θ)$ for $θ=−300°$ 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 $θ=−300°$ , so $2θ=−600°$ .

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

$f(−600°)=sin(−600°)$ ,

By the trigonometric property $sin(−θ)=−sinθ$ , therefore $sin(−600°)=−sin600°$ .

Therefore, $f(−600°)=−sin(600°)$

As $sin(360°+θ)=sinθ$ , substitute $θ=240°$ in the equation it gives $sin(360°+240°)=sin240°$ , it gives $sin600°=sin240°$ .

By the trigonometric identity $sin(180°+θ)=−sinθ$ , substitute $θ=60°$ it gives $sin(240°)=−sin60°$ .

Now substitute $32$ for $sin60°$ it gives $sin(600°)=−32$

Therefore value $f(2θ)$ for $θ=−300°$ is $−32$ .

(f)

To determine

To calculate: The exact value of function $g(−θ)$ for $θ=−300°$ , where $g(θ)=cosθ$ .

(f)

Expert Solution

Answer to Problem 114E

The value $g(−300°)$ is $12$ .

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 $θ=−300°$ in the equation it gives $g(−θ)=g(300°)=cos(300°)$ .

As per trigonometric identity $cos(360°−θ)=cosθ$ , where $θ$ is any acute angle.

Now substitute $θ=60°$ in the above equation it gives $cos(360°−60°)=cos60°$ , which gives $cos(300°)=cos60°$

Substitute $12$ for $cos(60°)$ it gives

$cos(300°)=12$ .

Therefore the value of $cos(−θ)$ for $θ=−300°$ is $12$ .

Chapter 4.4, Problem 113E

Chapter 4.4, Problem 115E

Chapter 4 Solutions

Precalculus with Limits: A Graphing Approach

Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - Prob. 7E Ch. 4.1 - Prob. 8E Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

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

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