Pregunta

What is the definition of arcsine in relation to sine?

Ask by Griffiths Tran. in South Africa
Jan 30,2025

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Arcsine, denoted as arcsin or sin⁻¹, is the inverse of the sine function. It takes a value between -1 and 1 and returns the angle between -90° and 90° whose sine is that value.

Solución

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:
In other words:

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:
Then:
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.

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Mind Expander

The arcsine function, often denoted as sin⁻¹(x), is the inverse of the sine function. It takes a value (within the range of -1 to 1) and returns the angle whose sine is that value, typically expressed in radians. For example, if sin(θ) = x, then arcsin(x) = θ, where θ is in the interval [-π/2, π/2]. This means that arcsine helps us find the angle when the sine value is known!
In practical scenarios, the arcsine function is widely used in fields such as physics and engineering, especially when dealing with waveforms, oscillations, or any situation involving angles and their sine values. For example, if you need to determine the angle of elevation based on a measured height and distance, calculating the arcsine provides a quick answer, making it a handy tool in your mathematical toolbox!

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