EBK PRECALCULUS W/LIMITS

4th Edition

ISBN: 9781337516853

Author: Larson

Publisher: CENGAGE CO

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Question

Chapter 5.3, Problem 79E

(a)

To determine

To find : the maximum and minimum point of the function

(a)

Expert Solution

Answer to Problem 79E

The maximum points of the function are $(π/3, 1.25) & (5π/3, 1.25) _$ radians.

The minimum point of the function are $(0,1)&(π, -1) _$ radians.

Explanation of Solution

Given information : Function $f(x)=sin2x+cosx$ ; Trigonometric Equation $2sinxcosx−sinx=0$ ; $[0,2π)$

Concept Involved:

A maximum is a high point and a minimum is a low point over the given interval.

Graph:

EBK PRECALCULUS W/LIMITS, Chapter 5.3, Problem 79E

Interpretation:

The graph shows, at $x=π/3 &5π/3$ radians the graph has the maximum value of 1.25 and at $x=0 &π$ radians the graph has the minimum value of $1&−1$ respectively.

(b)

To determine

To find : all the solutions of the trigonometric equations in the given interval

(b)

Expert Solution

Answer to Problem 79E

The solution to the given trigonometric equation are $x=0, π/3, π, & 5π/3 _$ radians.

Explanation of Solution

Given information : Function $f(x)=sin2x+cosx$ ; Trigonometric Equation $2sinxcosx−sinx=0$ ; $[0,2π)$

Concept Involved:

Solution to a quadratic equation are the values of x that makes the equation TRUE.

To solve a trigonometric equation, use standard algebraic techniques (when possible) such as collecting like terms, extracting square roots, and factoring.

Our preliminary goal in solving a trigonometric equation is to isolate the trigonometric function on one side of the equation.

Calculation:

Factor the Greatest Common Factor of left side of the equation $2sinxcosx−sinx=0$

$sinx(2cosx−1)=0$

Using the zero factor property which states that if $a⋅b=0$ then $a=0 (or) b=0$ , we need to set each factor to zero.

$sinx=0$ ▶1^s^t equation

$2cosx−1=0$ ▶2ⁿ^d equation

Solving the 1^s^t equation and finding x values that makes it true in the interval $[0,2π)$

$x=sin−1(0)$

$x=0, π$

Solving the 2ⁿ^d equation and finding x values that makes it true in the interval $[0,2π)$

By adding 1 on both sides of the equation
By simplifying on both sides of the equation
By dividing 2 on both sides of the equation
By simplifying fraction on both sides of the equation

$2cosx−1=02cosx=12cosx2=12cosx=12x=cos−1(12)x=π3, 5π3$

Conclusion:

$x=0, π3, π, &5π3$ radians are the zero of the function and solution to the equation $2sinxcosx−sinx=0$ in the interval $[0,2π)$

Chapter 5.3, Problem 78E

Chapter 5.3, Problem 80E

Chapter 5 Solutions

EBK PRECALCULUS W/LIMITS

Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E

Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - Prob. 49E Ch. 5.1 - Prob. 50E Ch. 5.1 - Prob. 51E Ch. 5.1 - Prob. 52E Ch. 5.1 - Prob. 53E Ch. 5.1 - Prob. 54E Ch. 5.1 - Prob. 55E Ch. 5.1 - Prob. 56E Ch. 5.1 - Prob. 57E Ch. 5.1 - Prob. 58E Ch. 5.1 - Prob. 59E Ch. 5.1 - Prob. 60E Ch. 5.1 - 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