교과서 질문
Solve each equation for all exact solutions, in degrees.
2√3 cos (θ/2) = -3
34
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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 6.2.33
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
csc² θ ―2 cot θ = 0
검증된 단계별 안내1
Recall the Pythagorean identity relating cosecant and cotangent: \(\csc^{2} \theta = 1 + \cot^{2} \theta\).
Substitute \(\csc^{2} \theta\) in the equation \(\csc^{2} \theta - 2 \cot \theta = 0\) with \(1 + \cot^{2} \theta\) to get \(1 + \cot^{2} \theta - 2 \cot \theta = 0\).
Rewrite the equation as a quadratic in terms of \(\cot \theta\): \(\cot^{2} \theta - 2 \cot \theta + 1 = 0\).
Solve the quadratic equation for \(\cot \theta\). Since it is a perfect square, factor it as \((\cot \theta - 1)^2 = 0\), which gives \(\cot \theta = 1\).
Find all angles \(\theta\) in the interval \([0^\circ, 360^\circ)\) where \(\cot \theta = 1\). Use the definition \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and find the corresponding \(\theta\) values.
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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.
Reciprocal and Quotient Identities
Understanding that cosecant (csc θ) is the reciprocal of sine (sin θ), and cotangent (cot θ) is the quotient of cosine over sine (cos θ / sin θ), is essential. These identities allow rewriting the equation in terms of sine and cosine, simplifying the solving process.
추천 영상:
03:40
Quotients of Complex Numbers in Polar Form
Pythagorean Identity involving Cotangent and Cosecant
The identity csc² θ = 1 + cot² θ relates cosecant and cotangent functions. This identity helps transform the given equation into a quadratic form in cot θ, making it easier to solve for θ within the specified interval.
추천 영상:
6:25Pythagorean Identities
Solving Trigonometric Equations within a Given Interval
After simplifying, solving for θ requires finding all solutions in [0°, 360°). This involves considering the periodicity of trigonometric functions and using inverse trigonometric functions to find exact or approximate angle measures.
추천 영상:
4:34How to Solve Linear Trigonometric Equations
관련 실천
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Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
sin x = sin 2x
35
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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.
2 sin θ = 2 cos 2θ
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Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
cos 2x + cos x = 0
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Solve each equation for exact solutions.
tan⁻¹ x - tan⁻¹ (1/x ) = π/6
647
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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.
2 - sin 2θ = 4 sin 2θ
31
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