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 6.2.45
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
cot θ + 2 csc θ = 3
검증된 단계별 안내1
Start by expressing the given equation \(\cot \theta + 2 \csc \theta = 3\) in terms of sine and cosine functions. Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\).
Rewrite the equation using these definitions: \(\frac{\cos \theta}{\sin \theta} + 2 \cdot \frac{1}{\sin \theta} = 3\).
Combine the terms over a common denominator \(\sin \theta\): \(\frac{\cos \theta + 2}{\sin \theta} = 3\).
Multiply both sides of the equation by \(\sin \theta\) to eliminate the denominator, giving \(\cos \theta + 2 = 3 \sin \theta\).
Rearrange the equation to isolate terms and prepare for solving: \(\cos \theta - 3 \sin \theta = -2\). From here, consider using trigonometric identities or methods such as expressing the left side as a single sine or cosine function to find \(\theta\) within the interval \([0^\circ, 360^\circ)\).
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, identities relating cotangent and cosecant, such as cot θ = cos θ / sin θ and csc θ = 1 / sin θ, are essential to rewrite and simplify the equation for easier solving.
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 algebraic manipulation, substitution, and using inverse trigonometric functions to determine exact or approximate angle values.
추천 영상:
4:34How to Solve Linear Trigonometric Equations
Interval and Angle Measurement
The problem restricts solutions to the interval [0°, 360°), meaning all solutions must be found within one full rotation of the unit circle. Understanding how angles correspond to points on the unit circle and how to interpret solutions within this range is crucial for providing correct and complete answers.
추천 영상:
5:31Reference Angles on the Unit Circle
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