교과서 질문
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
tan² θ + 4 tan θ + 2 = 0
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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial12th EditionTrigonometryISBN: 9780136552161
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TOC.tableOfContents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
7장, 문제 6.2.37
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
sec² θ tan θ = 2 tan θ
검증된 단계별 안내1
Start by writing down the given equation: \(\sec^{2} \theta \tan \theta = 2 \tan \theta\).
Recognize that \(\tan \theta\) is a common factor on both sides. To simplify, subtract \(2 \tan \theta\) from both sides to get: \(\sec^{2} \theta \tan \theta - 2 \tan \theta = 0\).
Factor out \(\tan \theta\) from the left side: \(\tan \theta (\sec^{2} \theta - 2) = 0\).
Set each factor equal to zero to find possible solutions: (1) \(\tan \theta = 0\) and (2) \(\sec^{2} \theta - 2 = 0\).
For (2), rewrite \(\sec^{2} \theta\) in terms of \(\tan^{2} \theta\) using the identity \(\sec^{2} \theta = 1 + \tan^{2} \theta\), then solve for \(\tan \theta\). Finally, find all \(\theta\) in \([0^\circ, 360^\circ)\) that satisfy these conditions.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing that sec²θ = 1 + tan²θ helps simplify and solve the equation by expressing all terms in a single trigonometric function.
추천 영상:
5:32
Fundamental Trigonometric Identities
Solving Trigonometric Equations
Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a specified interval. This often requires factoring, using identities, and considering the periodic nature of functions like tangent and secant.
추천 영상:
4:34How to Solve Linear Trigonometric Equations
Interval and Solution Set for Angles
When solving trigonometric equations over [0°, 360°), it is essential to find all solutions within one full rotation of the unit circle. Understanding how to interpret and express solutions as exact values or decimal approximations ensures completeness and accuracy.
추천 영상:
04:46Coterminal Angles
관련 실천
교과서 질문
Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
3 csc² x/2 = 2 sec x
36
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교과서 질문
Solve for exact solutions over the interval [0°, 360°).
cos θ/2 = -1/2
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교과서 질문
Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
8 sec² x/2 = 4
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교과서 질문
The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval [0, 2π). Express solutions to four decimal places.
x² + sin x - x³ - cos x = 0
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교과서 질문
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
2 tan θ sin θ - tan θ = 0
33
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