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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 105
Capitolo 3, Problema 105

In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 2 sin² x = 3 - sin x

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Rewrite the given equation to standard form by bringing all terms to one side: \(2 \sin^{2} x + \sin x - 3 = 0\).
Recognize that this is a quadratic equation in terms of \(\sin x\). Let \(u = \sin x\), so the equation becomes \(2u^{2} + u - 3 = 0\).
Solve the quadratic equation \(2u^{2} + u - 3 = 0\) using the quadratic formula: \(u = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=2\), \(b=1\), and \(c=-3\).
Find the values of \(u\) (which represent \(\sin x\)) from the quadratic formula and determine which values are valid since \(\sin x\) must be in the interval \([-1, 1]\).
For each valid \(\sin x\) value, solve for \(x\) in the interval \([0, 2\pi)\) by using the inverse sine function and considering the sine function's symmetry to find all solutions.

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Solving Quadratic Trigonometric Equations

This involves rewriting the trigonometric equation in a quadratic form, such as ax² + bx + c = 0, by substituting expressions like sin²x with a variable. Once in quadratic form, standard algebraic methods like factoring or the quadratic formula can be applied to find solutions for the trigonometric function.
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Solving Quadratic Equations by Completing the Square

Unit Circle and Interval Restrictions

Understanding the unit circle is essential to interpret solutions for trigonometric functions within a specific interval, here [0, 2π). This helps identify all possible angles that satisfy the equation, considering the periodic nature of sine and ensuring solutions fall within the given domain.
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Introduction to the Unit Circle

Exact Values and Approximate Solutions

Some trigonometric equations yield solutions with well-known exact values (like π/6 or π/4), while others require numerical approximation. Knowing when and how to provide exact values or approximate decimal answers (to four decimal places) is crucial for correctly presenting solutions.
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Example 1
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