Here are the essential concepts you must grasp in order to answer the question correctly.
Quadratic Equations in Trigonometry
The given equation, 2 sinΒ² x + sin x - 2 = 0, is a quadratic equation in terms of sin x. Quadratic equations can be solved using factoring, the quadratic formula, or completing the square. Understanding how to manipulate and solve these equations is essential for finding the values of x that satisfy the equation within the specified interval.
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Trigonometric Functions and Their Values
Trigonometric functions, such as sine, have specific values at key angles (e.g., 0, Ο/2, Ο, 3Ο/2, and 2Ο). Knowing the unit circle and the values of sin x at these angles is crucial for solving trigonometric equations. This knowledge allows for the identification of exact solutions and helps in approximating values when necessary.
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Interval Notation and Solutions
The interval [0, 2Ο) indicates that solutions must be found within one full rotation of the unit circle, from 0 to just below 2Ο. Understanding how to interpret and apply interval notation is important for determining valid solutions. It ensures that all solutions are within the specified range and helps in identifying any periodic nature of the trigonometric functions involved.
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