Precalculus: Mathematics for Calculus - 6th Edition
Precalculus: Mathematics for Calculus - 6th Edition
6th Edition
ISBN: 9780840068071
Author: Stewart, James, Redlin, Lothar, Watson, Saleem
Publisher: Cengage Learning
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Chapter 7.5, Problem 5E
To determine

To solve : the given equation { tan2θ2secθ=2 .

Expert Solution & Answer
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Answer to Problem 5E

The general solutions of θ for the equation { tan2θ2secθ=2 are

  θ=1.231+2nπ , θ=5.052+2nπ   or   θ=π+2nπ .

Explanation of Solution

Given information:

The equation to be solved is { tan2θ2secθ=2 .

Calculation :

  tan2θ2secθ=2............(1)

The trigonometric identity is {tan2θ=sec2θ1 .

Put the value of tan2θ in equation 1.

  sec2θ12secθ=2sec2θ2secθ3=0

Put secθ=x .

  x22x3=0

Factorize the above equation.

  x23x+x3=0x(x3)+1(x3)=0(x+1)(x3)=0x=1    or     x=3

Or

  secθ=1   or   secθ=3orcosθ=1   or  cosθ=13

Now, there are two cases. Solve each case to get the solution of θ .

Case 1:

  cosθ=1

Above equation is satisfied in the interval [0,2π] when θ=π .

Since, the period of cosine function is 2π . Add 2πn to get the general solution of θ where n is any integer.

  θ=π+2nπ

Case 2:

  cosθ=13

Above equation is satisfied in the interval [0,2π] .

Use calculator to solve cosθ=13 .

  θ=1.231 radian  or  θ=5.052 radian

Since, the period of cosine function is 2π . Add 2πn to get the general solution of θ where n is any integer.

  θ=1.231+2nπ  or  θ=5.052+2nπ

Therefore, the general solutions of θ for the equation { tan2θ2secθ=2 are

  θ=1.231+2nπ , θ=5.052+2nπ   or   θ=π+2nπ .

Chapter 7 Solutions

Precalculus: Mathematics for Calculus - 6th Edition

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Prob. 60ECh. 7.5 - Prob. 61ECh. 7.5 - Prob. 62ECh. 7.5 - Prob. 63ECh. 7.5 - Damped Vibrations The displacement of a spring...Ch. 7.5 - Hours of Daylight In Philadelphia the number of...Ch. 7.5 - Belts and Pulleys A thin belt of length L...Ch. 7.5 - Prob. 67ECh. 7 - Prob. 1RCCCh. 7 - Prob. 2RCCCh. 7 - Prob. 3RCCCh. 7 - Prob. 4RCCCh. 7 - Prob. 5RCCCh. 7 - Prob. 6RCCCh. 7 - Prob. 7RCCCh. 7 - Prob. 8RCCCh. 7 - Prob. 1RECh. 7 - Prob. 2RECh. 7 - Prob. 3RECh. 7 - Prob. 4RECh. 7 - Prob. 5RECh. 7 - Prob. 6RECh. 7 - Prob. 7RECh. 7 - Prob. 8RECh. 7 - Prob. 9RECh. 7 - Prob. 10RECh. 7 - Prob. 11RECh. 7 - Prob. 12RECh. 7 - Prob. 13RECh. 7 - Prob. 14RECh. 7 - Prob. 15RECh. 7 - Prob. 16RECh. 7 - Prob. 17RECh. 7 - Prob. 18RECh. 7 - Prob. 19RECh. 7 - Prob. 20RECh. 7 - Prob. 21RECh. 7 - Prob. 22RECh. 7 - Prob. 23RECh. 7 - Prob. 24RECh. 7 - Prob. 25RECh. 7 - Prob. 26RECh. 7 - Prob. 27RECh. 7 - Prob. 28RECh. 7 - Prob. 29RECh. 7 - Prob. 30RECh. 7 - Prob. 31RECh. 7 - Prob. 32RECh. 7 - Prob. 33RECh. 7 - Prob. 34RECh. 7 - 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(a) sin 8...Ch. 7 - For the angles and in the figures, find cos( +...Ch. 7 - Prob. 5TCh. 7 - Prob. 6TCh. 7 - Prob. 7TCh. 7 - Prob. 8TCh. 7 - Find the exact value of cos(2tan1940).Ch. 7 - Rewrite the expression as an algebraic function of...Ch. 7 - Wave on a Canal A wave on the surface of a long...Ch. 7 - Prob. 2PCh. 7 - Traveling Wave A traveling wave is graphed at the...Ch. 7 - Traveling Wave A traveling wave has period 2/3,...Ch. 7 - Standing Wave A standing wave with amplitude 0.6...Ch. 7 - Prob. 6PCh. 7 - Prob. 7PCh. 7 - Prob. 8PCh. 7 - Prob. 1CRTCh. 7 - Prob. 2CRTCh. 7 - Prob. 3CRTCh. 7 - Prob. 4CRTCh. 7 - Prob. 5CRTCh. 7 - Prob. 6CRTCh. 7 - Prob. 7CRTCh. 7 - Prob. 8CRTCh. 7 - Prob. 9CRTCh. 7 - Prob. 10CRTCh. 7 - Prob. 11CRTCh. 7 - Prob. 12CRT
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