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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 21
Capitolo 3, Problema 21

Find all solutions of each equation. 4 sin θ﹣1 = 2 sin θ

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Start by rewriting the given equation: \(4 \sin \theta - 1 = 2 \sin \theta\).
Bring all terms involving \(\sin \theta\) to one side to isolate the trigonometric function: \(4 \sin \theta - 2 \sin \theta = 1\).
Simplify the left side: \(2 \sin \theta = 1\).
Solve for \(\sin \theta\) by dividing both sides by 2: \(\sin \theta = \frac{1}{2}\).
Find all angles \(\theta\) where \(\sin \theta = \frac{1}{2}\), considering the domain of \(\theta\) (usually \(0^\circ\) to \(360^\circ\) or \(0\) to \(2\pi\) radians), and use the unit circle or inverse sine function to determine these solutions.

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Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle values that satisfy the equation within a given domain. This often requires algebraic manipulation and understanding the periodic nature of sine, cosine, or other trig functions.
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Properties of the Sine Function

The sine function, sin θ, is periodic with period 2π and ranges between -1 and 1. Knowing its values and symmetry helps find all solutions to equations involving sine, including using reference angles and considering all quadrants where sine has the required value.
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Graph of Sine and Cosine Function

Algebraic Manipulation of Trigonometric Equations

Rearranging and simplifying equations like 4 sin θ - 1 = 2 sin θ requires combining like terms and isolating sin θ. This step is crucial before applying inverse trigonometric functions to find angle solutions.
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