교과서 질문
In Exercises 67–74, rewrite each expression in terms of the given function or functions. ;
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
3장, 문제 69
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). sin 2x = cos x
검증된 단계별 안내1
Recall the double-angle identity for sine: \(\sin 2x = 2 \sin x \cos x\). Substitute this into the equation to get \(2 \sin x \cos x = \cos x\).
Rewrite the equation as \(2 \sin x \cos x - \cos x = 0\) to set it equal to zero, which allows factoring.
Factor out \(\cos x\) from the left side: \(\cos x (2 \sin x - 1) = 0\).
Set each factor equal to zero to find possible solutions: \(\cos x = 0\) and \(2 \sin x - 1 = 0\).
Solve each equation separately on the interval \([0, 2\pi)\): For \(\cos x = 0\), find all \(x\) where cosine is zero; for \(2 \sin x - 1 = 0\), solve for \(\sin x = \frac{1}{2}\) and find all corresponding \(x\) values.
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주요 개념
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, the double-angle identity for sine, sin(2x) = 2 sin x cos x, is essential to rewrite the equation in a solvable form.
추천 영상:
5:32
Fundamental Trigonometric Identities
Solving Trigonometric Equations
Solving trigonometric equations involves manipulating the equation using identities and algebraic techniques to isolate the variable. After applying identities, solutions are found by setting factors equal to zero or using inverse trigonometric functions within the given interval.
추천 영상:
4:34How to Solve Linear Trigonometric Equations
Interval Restriction and General Solutions
When solving trigonometric equations, solutions must be restricted to the specified interval, here [0, 2π). This means finding all angles within one full rotation of the unit circle that satisfy the equation, considering periodicity and multiple solutions.
추천 영상:
4:34How to Solve Linear Trigonometric Equations
관련 실천
교과서 질문
In Exercises 69–74, rewrite each expression as a simplified expression containing one term. sin (α - β) cos β + cos (α - β) sin β
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교과서 질문
In Exercises 54–67, 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 + sin x - 2 = 0
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교과서 질문
In Exercises 67–74, rewrite each expression in terms of the given function or functions. ;
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교과서 질문
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 4 cos² x = 5 - 4 sin x
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교과서 질문
In Exercises 69–74, rewrite each expression as a simplified expression containing one term. cos (α + β) cos β + sin (α + β) sin β
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