What is the definition of arcsine in relation to sine?

The **arcsine** function, often denoted as **arcsin** or **sin⁻¹**, is the inverse operation of the **sine** function. Understanding the relationship between these two functions involves recognizing how they correspond to each other within specific domains and ranges. ### **Definition of Arcsine in Relation to Sine** - **Sine Function (sin):** - **Domain:** All real numbers (−∞, ∞) - **Range:** [−1, 1] - **Behavior:** Periodic and oscillates between −1 and 1. - **Arcsine Function (arcsin or sin⁻¹):** - **Domain:** [−1, 1] *(since sine values are confined within this interval)* - **Range:** [−π/2, π/2] or [−90°, 90°] *(principal values)* - **Behavior:** Provides the angle whose sine is a given number within the specified domain and range. ### **Formal Definition** For a real number **x** within the interval [−1, 1], the arcsine of **x** is the unique angle **y** in the interval [−π/2, π/2] (or [−90°, 90°]) such that: \[ \sin(y) = x \quad \text{and} \quad y = \arcsin(x) \] In other words: \[ \arcsin\left(\sin(y)\right) = y \quad \text{for} \quad y \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \] \[ \sin\left(\arcsin(x)\right) = x \quad \text{for} \quad x \in [-1, 1] \] ### **Why Restrict the Domain?** The sine function is **not one-to-one** over its entire domain because it is periodic, meaning it repeats its values in regular intervals. To define an inverse function like arcsine, we must **restrict the domain** of the sine function to an interval where it is one-to-one. The interval [−π/2, π/2] is chosen for this purpose because, within this range, each sine value corresponds to exactly one angle, making the inverse function well-defined. ### **Graphical Interpretation** On a graph: - The **sine function** curves up and down, crossing the x-axis at multiples of π. - The **arcsine function** is the inverse, reflecting the sine curve over the line y = x but limited to the principal domain and range to maintain its function properties. ### **Practical Example** If you know that: \[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \] Then: \[ \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6} \] This means that the arcsine function takes the sine value of 1/2 and returns the angle π/6 radians (or 30 degrees) within the principal range. ### **Summary** The **arcsine** function serves as the inverse of the sine function by mapping a sine value back to its corresponding angle within a specific interval. This relationship is fundamental in trigonometry, especially when solving for angles in various applications such as physics, engineering, and geometry.