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Ch. 6 - Inverse Circular Functions and Trigonometric Equations
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 6 - Inverse Circular Functions and Trigonometric EquationsProblem 6.2.45
Chapter 7, Problem 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

Verified step by step guidance
1
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 \([0^\circ, 360^\circ)\).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.

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Textbook Question

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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.


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