교과서 질문
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 4 tan² x - 8 tan x + 3 = 0
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
3장, 문제 99
In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x + sin x = 0
검증된 단계별 안내1
Start by recognizing the given equation: \(\sin 2x + \sin x = 0\). Our goal is to solve for \(x\) in the interval \([0, 2\pi)\).
Use the double-angle identity for sine: \(\sin 2x = 2 \sin x \cos x\). Substitute this into the equation to get \(2 \sin x \cos x + \sin x = 0\).
Factor out the common factor \(\sin x\): \(\sin x (2 \cos x + 1) = 0\). This product equals zero, so set each factor equal to zero separately.
Solve the first equation \(\sin x = 0\) for \(x\) in \([0, 2\pi)\), and then solve the second equation \(2 \cos x + 1 = 0\) for \(x\) in the same interval.
For \(2 \cos x + 1 = 0\), isolate \(\cos x\) to get \(\cos x = -\frac{1}{2}\). Find all \(x\) in \([0, 2\pi)\) where \(\cos x = -\frac{1}{2}\). Combine these solutions with those from \(\sin x = 0\) to get the complete solution set.
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주요 개념
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Double-Angle Identity for Sine
The double-angle identity expresses sin(2x) as 2 sin(x) cos(x). This allows rewriting the equation sin 2x + sin x = 0 into a form involving sin(x) and cos(x), facilitating factorization and solution. Recognizing and applying this identity simplifies trigonometric equations.
추천 영상:
05:06
Double Angle Identities
Solving Trigonometric Equations Using Factoring
Factoring trigonometric expressions helps break down complex equations into simpler parts. After rewriting sin 2x + sin x = 0 using identities, factoring out common terms like sin(x) can isolate factors, each set to zero, yielding possible solutions within the given interval.
추천 영상:
04:42Solve Trig Equations Using Identity Substitutions
Finding Solutions on a Specific Interval
Trigonometric equations often have multiple solutions within a given interval, such as [0, 2π). It is essential to find all solutions that satisfy the equation in this range, using exact values when possible or approximations rounded to four decimal places, ensuring completeness and accuracy.
추천 영상:
3:43Finding the Domain of an Equation
관련 실천
교과서 질문
In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. cos x - 5 = 3 cos x + 6
487
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교과서 질문
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 7 sin² x - 1 = 0
497
views
교과서 질문
In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 cos 2x + 1 = 0
557
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교과서 질문
In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 sin² x = 3 - sin x
519
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교과서 질문
In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. tan x sec x = 2 tan x
536
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