Ch. 3 - Trigonometric Identities and Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 3 - Trigonometric Identities and Equations

Problem 21

Chapter 3, Problem 21

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

Verified step by step guidance

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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Key Concepts

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

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.

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.

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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