Textbook Question
Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth, as appropriate.
sin² θ ― 2 sin θ + 3 = 0
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6. Trigonometric Identities and More Equations
Solving Trigonometric Equations Using Identities
Problem 6.2.55
Textbook Question
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.
sin θ cos θ ― sin θ = 0
Verified step by step guidance1
Start by writing down the given equation: \(\sin \theta \cos \theta - \sin \theta = 0\).
Factor out the common factor \(\sin \theta\) from the left-hand side: \(\sin \theta (\cos \theta - 1) = 0\).
Set each factor equal to zero to find possible solutions: \(\sin \theta = 0\) or \(\cos \theta - 1 = 0\).
Solve \(\sin \theta = 0\) for \(\theta\) in degrees within the range \(0^\circ \leq \theta < 360^\circ\). Recall that \(\sin \theta = 0\) at \(\theta = 0^\circ, 180^\circ, 360^\circ\).
Solve \(\cos \theta - 1 = 0\) which simplifies to \(\cos \theta = 1\). Find \(\theta\) in degrees where this is true within \(0^\circ \leq \theta < 360^\circ\). Remember that \(\cos \theta = 1\) at \(\theta = 0^\circ\).
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Trigonometric Equations
Solving trigonometric equations involves finding all angle values that satisfy the given equation within a specified domain. This often requires factoring, using identities, or isolating trigonometric functions. Solutions may be exact (in terms of π or degrees) or approximate, depending on the problem.
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Trigonometric Identities and Factoring
Using identities like the distributive property and factoring is essential to simplify and solve equations such as sin θ cos θ - sin θ = 0. Factoring out common terms (e.g., sin θ) helps break the equation into simpler parts, allowing separate solutions for each factor.
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Angle Measures and Solution Sets
Understanding angle measures in radians and degrees is crucial for interpreting solutions correctly. The problem requires expressing answers as least nonnegative angles and rounding appropriately, which involves converting between radians and degrees and considering periodicity to find all valid solutions.
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