Understanding Inverse Sine (Arcsin)
The inverse sine function, written as arcsin(x) or sin-1(x), returns the angle whose sine is x. If sin(θ) = x, then arcsin(x) = θ. The function is defined for inputs in [-1, 1] and produces outputs in [-90°, 90°] or [-π/2, π/2] radians.
Common Arcsin Values
arcsin(0) = 0°
The sine of 0 degrees is 0, so arcsin(0) returns 0 degrees.
arcsin(1/2) = 30°
One of the most fundamental values from the 30-60-90 triangle.
arcsin(√2/2) = 45°
From the 45-45-90 isosceles right triangle on the unit circle.
arcsin(√3/2) = 60°
The complement of 30 degrees in a 30-60-90 triangle.
arcsin(1) = 90°
The maximum output angle. sin(90°) = 1.
arcsin(-1) = -90°
The minimum output angle. sin(-90°) = -1.
Domain and Range
The domain of arcsin is [-1, 1] and the range is [-π/2, π/2] or [-90°, 90°]. This restriction ensures the function is one-to-one, producing unique outputs. The range covers quadrants I and IV on the unit circle.
Key Properties
- arcsin(x) is an increasing function: as x increases, the angle increases.
- arcsin(x) + arccos(x) = π/2 for all x in [-1, 1].
- sin(arcsin(x)) = x for all x in [-1, 1].
- arcsin(-x) = -arcsin(x) (odd function, symmetric about origin).
- The derivative: d/dx arcsin(x) = 1/√(1 - x²).
- arcsin(0) = 0 (passes through the origin).
Relationship with Other Inverse Trig Functions
- arcsin(x) = π/2 - arccos(x) (complementary relationship).
- arcsin(x) = arctan(x/√(1 - x²)) for |x| < 1.
- arcsin(x) = 2 arctan(x/(1 + √(1 - x²))).
Applications of Inverse Sine
- Physics: Finding launch angles in projectile motion problems.
- Engineering: Calculating angles of elevation and depression.
- Navigation: Determining latitude from stellar observations.
- Optics: Snell's law uses arcsin to find refraction angles.
- Signal processing: Phase angle calculations.