Find the exact value of s in the given interval that has the given circular function value.


[π/2, π] ; sin s = 1/2

Verified step by step guidance

1. Identify the interval [\(\frac{\pi}{2}, \pi\)] which corresponds to the second quadrant of the unit circle.
2. Recall that in the second quadrant, the sine function is positive.
3. Recognize that \(\sin s = \frac{1}{2}\) is a common angle value, typically associated with \(s = \frac{\pi}{6}\) or \(s = \frac{5\pi}{6}\).
4. Since \(\frac{\pi}{6}\) is not in the interval [\(\frac{\pi}{2}, \pi\)], consider the angle \(s = \frac{5\pi}{6}\) which is in the second quadrant.
5. Verify that \(\sin(\frac{5\pi}{6}) = \frac{1}{2}\) to ensure the solution is correct.

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric interpretation of the sine, cosine, and tangent functions. The coordinates of points on the unit circle correspond to the cosine and sine values of angles, allowing for the determination of trigonometric function values for various angles.

Sine Function

The sine function, denoted as sin(θ), represents the ratio of the length of the opposite side to the hypotenuse in a right triangle. In the context of the unit circle, it gives the y-coordinate of a point on the circle corresponding to an angle θ. Understanding the sine function is crucial for solving equations involving sine values, such as finding angles that yield specific sine outputs.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, are used to find angles when the value of a trigonometric function is known. For example, if sin(s) = 1/2, then s can be determined using arcsin(1/2). It is important to consider the range of the inverse functions and the specified interval when solving for angles, as multiple angles can yield the same sine value.