EBK PRECALCULUS W/LIMITS

4th Edition

ISBN: 9781337516853

Author: Larson

Publisher: CENGAGE CO

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is…

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Question

Chapter 5.3, Problem 82E

(a)

To determine

To find : the maximum and minimum point of the function

(a)

Expert Solution

Answer to Problem 82E

The maximum points of the function are (π/6, 1.5) & (5π/6, 1.5) _ radians.

The minimum point of the function are (π/2, 1) & (3π/2, -3) _ radians.

Explanation of Solution

Given information : Function f(x)=2sinx+cos2x ; Trigonometric Equation 2cosx−4sinxcosx=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 82E

Interpretation:

The graph of the function f(x)=2sinx+cos2x shows, at x=π6, 5π6 radians the graph has the maximum value of 1.5 and at x=π2, 3π2 radians the graph has the minimum value of 1 & −3 respectively.

(b)

To determine

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

(b)

Expert Solution

Answer to Problem 82E

The solution to the given trigonometric equation are x= π6, π2, 5π6,& 3π2 _ radians.

Explanation of Solution

Given information : Function f(x)=2sinx+cos2x ; Trigonometric Equation 2cosx−4sinxcosx=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 from left side of the equation 2cosx−4sinxcosx=0

2cosx(1−2sinx)=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.

2cosx=0▶1^s^t equation

1−2sinx=0▶2ⁿ^d equation

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

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

2cosx2=02cosx=0x=cos−1(0)x=π2, 3π2

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

By subtracting1 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

1−2sinx=01−2sinx−1=0−1−2sinx=−1−2sinx−2=−1−2sinx=12x=sin−1(12)x=π6, 5π6

Conclusion:

x=π2, 3π2, π6, & 5π6 radians are the zero of the function and solution to the equation 2cosx−4sinxcosx=0 in the interval [0,2π)

Chapter 5.3, Problem 81E

Chapter 5.3, Problem 83E

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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