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 67
In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). 4 cos² x = 5 - 4 sin x
검증된 단계별 안내1
Start by recalling the Pythagorean identity: \(\cos^{2} x = 1 - \sin^{2} x\). This allows us to rewrite the equation in terms of \(\sin x\) only.
Substitute \(\cos^{2} x\) with \(1 - \sin^{2} x\) in the equation: \(4(1 - \sin^{2} x) = 5 - 4 \sin x\).
Expand the left side: \(4 - 4 \sin^{2} x = 5 - 4 \sin x\).
Bring all terms to one side to form a quadratic equation in \(\sin x\): \(-4 \sin^{2} x + 4 \sin x + 4 - 5 = 0\), which simplifies to \(-4 \sin^{2} x + 4 \sin x - 1 = 0\).
Multiply the entire equation by \(-1\) to make the quadratic coefficient positive: \(4 \sin^{2} x - 4 \sin x + 1 = 0\). Now, solve this quadratic equation for \(\sin x\) using the quadratic formula.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Pythagorean Identity
The Pythagorean identity states that sin²x + cos²x = 1 for any angle x. This identity allows you to express cos²x in terms of sin²x or vice versa, which is essential for rewriting the given equation into a single trigonometric function to simplify solving.
추천 영상:
6:25
Pythagorean Identities
Solving Trigonometric Equations
Solving trigonometric equations involves manipulating the equation using identities and algebraic techniques to isolate the trigonometric function. Then, find all angle solutions within the specified interval by considering the unit circle and periodicity of sine and cosine.
추천 영상:
4:34How to Solve Linear Trigonometric Equations
Interval Restriction and General Solutions
When solving trigonometric equations, solutions must be found within a given interval, here [0, 2π). This requires understanding the unit circle and how sine and cosine values repeat every 2π, ensuring all valid solutions in the interval are identified and extraneous ones discarded.
추천 영상:
4:34How to Solve Linear Trigonometric Equations
관련 실천
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교과서 질문
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