In Exercises 99–104, find two values of θ, 0 ≤ θ < 2𝜋, that satisfy each equation._sin θ = √22

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Recognize that the equation \( \sin \theta = \frac{\sqrt{2}}{2} \) corresponds to the sine values of special angles in the unit circle.

Recall that \( \sin \theta = \frac{\sqrt{2}}{2} \) at angles \( \theta = \frac{\pi}{4} \) and \( \theta = \frac{3\pi}{4} \) in the interval \( 0 \leq \theta < 2\pi \).

Verify that these angles are within the specified range \( 0 \leq \theta < 2\pi \).

Consider the periodic nature of the sine function, which repeats every \( 2\pi \), to ensure no additional solutions exist within the given interval.

Conclude that the two values of \( \theta \) that satisfy the equation are \( \frac{\pi}{4} \) and \( \frac{3\pi}{4} \).

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Key Concepts

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

Sine Function

The sine function, denoted as sin(θ), is a fundamental trigonometric function that relates the angle θ in a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is periodic with a range of [-1, 1], meaning it can only take values within this interval. Understanding the sine function is crucial for solving equations involving sin(θ), as it helps identify possible angles that yield specific sine values.

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a vital tool in trigonometry for visualizing the values of sine and cosine for various angles. By using the unit circle, one can determine the angles that correspond to specific sine values, such as √2/2, and find all possible angles within the specified range of 0 to 2π.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, are used to find the angle corresponding to a given sine value. For example, if sin(θ) = √2/2, the arcsin function can help identify the principal angle. However, since sine is positive in both the first and second quadrants, it is essential to find all angles that satisfy the equation within the specified interval, which may involve adding π to the principal angle to find the second solution.

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