Definitions and Graphs
Inverse trigonometric functions are the reverse operations of the standard trigonometric functions. While sin(θ) takes an angle and returns a ratio, arcsin(x) takes a ratio and returns the angle that produces it. These functions are essential in solving equations like sin(θ) = 0.5, where we need to find the unknown angle rather than an unknown ratio. Because trigonometric functions are periodic and therefore not one-to-one over their entire domains, we must restrict their domains before defining proper inverses. For example, sine repeats every 2π radians, so we restrict it to [−π/2, π/2] to create a one-to-one function whose inverse, arcsin, is well-defined. Each inverse trig function has a carefully chosen principal value range that makes it a true function. The six inverse trigonometric functions are arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹), arccot (cot⁻¹), arcsec (sec⁻¹), and arccsc (csc⁻¹). At the high school level, the three most commonly used are arcsin, arccos, and arctan. Understanding their domains, ranges, and graphs is fundamental to working with angles in physics, engineering, and advanced mathematics.
- Why We Must Restrict Domains
- arcsin(x) — Inverse Sine
- arccos(x) — Inverse Cosine
- arctan(x) — Inverse Tangent
- The Fundamental Inverse Relationship
Why We Must Restrict Domains
A function must be one-to-one (injective) to have an inverse. The sine function, for instance, gives sin(π/6) = sin(5π/6) = 0.5, so two different inputs produce the same output. If we tried to define arcsin(0.5), there would be infinitely many valid answers. To fix this, mathematicians agree on a restricted domain for each trig function that keeps it one-to-one, and the corresponding inverse is called the principal value.
arcsin(x) — Inverse Sine
We restrict sin to the domain [−π/2, π/2], where it is strictly increasing. The inverse is y = arcsin(x), defined for x ∈ [−1, 1] with range [−π/2, π/2]. Key values: arcsin(0) = 0, arcsin(1) = π/2, arcsin(−1) = −π/2, arcsin(1/2) = π/6. The graph passes through the origin, is increasing, and has a characteristic S-shape that is the reflection of the restricted sine curve across the line y = x.
arccos(x) — Inverse Cosine
We restrict cos to the domain [0, π], where it is strictly decreasing. The inverse is y = arccos(x), defined for x ∈ [−1, 1] with range [0, π]. Key values: arccos(1) = 0, arccos(0) = π/2, arccos(−1) = π, arccos(1/2) = π/3. Note that arcsin(x) + arccos(x) = π/2 for all x ∈ [−1, 1], a useful identity. The graph is decreasing and also S-shaped, reflected from the restricted cosine graph.
arctan(x) — Inverse Tangent
We restrict tan to the open interval (−π/2, π/2), where it is strictly increasing with vertical asymptotes at the endpoints. The inverse is y = arctan(x), defined for all real numbers x ∈ (−∞, ∞) with range (−π/2, π/2). Key values: arctan(0) = 0, arctan(1) = π/4, arctan(−1) = −π/4. The graph has horizontal asymptotes at y = π/2 and y = −π/2, approaching but never reaching them as x → ±∞.
The Fundamental Inverse Relationship
For each inverse function, the composition laws hold only within the restricted domain. Specifically: sin(arcsin(x)) = x for x ∈ [−1, 1], but arcsin(sin(x)) = x only if x ∈ [−π/2, π/2]. For example, arcsin(sin(5π/6)) = arcsin(1/2) = π/6, not 5π/6, because π/6 is the principal value. Always check whether the input is within the principal value range before applying cancellation.
The Key Equation
- x — Input (ratio) (dimensionless): The value of the sine ratio, always between −1 and 1 inclusive.
- y — Output (angle) (radians): The principal angle whose sine equals x, restricted to [−π/2, π/2].
- \arcsin — Inverse sine function (radians output): Also written sin⁻¹; maps a ratio back to its principal angle.
Worked Example
Problem. A right triangle has an opposite side of length 7 and a hypotenuse of length 10. Find the angle θ (in degrees and radians) opposite the side of length 7, using the inverse sine function.
Answer. θ ≈ 44.43° ≈ 0.7754 radians
Origins · 1 minDaniel Bernoulli and how this was discovered
While Euler systematized the notation for trigonometric functions, it was mathematicians of the Bernoulli family and their contemporaries who explored inverse relationships and integral representations. Daniel Bernoulli contributed to the understanding of trigonometric series and the physical problems — such as oscillating strings — that make inverse trig functions indispensable. The notation 'arcsin' was popularized in the 18th century partly through the work of this era's analysts. Daniel Bernoulli was one of at least eight prominent mathematicians in the Bernoulli family spanning three generations, and his father Johann once tried to disinherit him after they tied for first place in a prestigious Paris Academy prize.
Related Concepts
- Definitions of Sine, Cosine, and Tangent — Inverse trig functions are defined by reversing sine, cosine, and tangent; you must know the forward functions to understand their inverses.
- The Unit Circle — The unit circle provides the coordinate pairs that define sine and cosine values, directly informing the key values and ranges of arcsin and arccos.
- Inverse Functions and Their Notation — Inverse trig functions are a specific application of the general concept of inverse functions; understanding domain restrictions and the horizontal line test is essential.
- Trigonometric Identities — Identities such as arcsin(x) + arccos(x) = π/2 connect inverse trig functions and are used in simplification and integration.
- Solving Trigonometric Equations — Inverse trig functions are the primary tool for isolating angle variables when solving equations like sin(θ) = 0.8.