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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Key Concepts

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

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.

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.

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.