inverse sin of 1/2

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30-03-2025

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The inverse sine function, also known as arcsine or sin⁻¹, answers the question: "What angle has a sine value of x?" In this case, we're looking for the inverse sine of 1/2, or sin⁻¹(1/2). This means we need to find the angle whose sine is equal to 1/2.

Finding the Principal Value

The sine function is periodic, meaning it repeats its values at regular intervals. This means there are infinitely many angles whose sine is 1/2. However, the inverse sine function, by convention, only returns a single value – the principal value. This value is typically restricted to a specific range: -π/2 ≤ x ≤ π/2 (or -90° ≤ x ≤ 90° in degrees).

To find the principal value of sin⁻¹(1/2), we can consider the unit circle or the sine graph.

Using the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Points on the unit circle are represented by (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis.

We are looking for an angle θ where sin θ = 1/2. Looking at the unit circle, we see that this occurs at two points. However, only one of these falls within the range of the principal value (-π/2 to π/2). That angle is π/6 or 30°.

Using the Sine Graph

The graph of y = sin(x) shows the sine value for various angles. We're looking for the point where y = 1/2. You'll find that this happens at multiple points, but the principal value—the value between -π/2 and π/2—is again π/6.

Other Solutions

While π/6 is the principal value, remember that the sine function is periodic with a period of 2π. Therefore, there are infinitely many angles whose sine is 1/2. These can be expressed as:

• π/6 + 2kπ, where k is any integer.
• 5π/6 + 2kπ, where k is any integer.

The second solution, 5π/6, is outside the principal value range of the inverse sine function.

Applications of Inverse Sine

The inverse sine function has many applications in various fields, including:

• Trigonometry: Solving triangles, finding angles given side lengths.
• Physics: Calculating angles of projectile motion, analyzing wave phenomena.
• Engineering: Designing structures, analyzing mechanical systems.
• Computer graphics: Calculating transformations and rotations.

Conclusion

The inverse sine of 1/2, or sin⁻¹(1/2), has a principal value of π/6 radians (or 30 degrees). While there are infinitely many angles whose sine is 1/2, the principal value is the only one returned by the inverse sine function. Understanding this concept is crucial for working with trigonometric functions and their applications.