Enter sine, cosine, or tangent values to compute arcsin / arccos / arctan — results in both radians & degrees
arcsin (inverse sine)
sin⁻¹()
arccos (inverse cosine)
cos⁻¹()
arctan (inverse tangent)
tan⁻¹()
Radians ⇄ Degrees conversion:rad =≈ 45°
Results
Step-by-Step Derivation
Inverse Trigonometric Formulas
arcsin(x) ∈ [-π/2, π/2] inverse sine, range -90° to 90°
arccos(x) ∈ [0, π] inverse cosine, range 0° to 180°
arctan(x) ∈ (-π/2, π/2) inverse tangent, range -90° to 90°
Radians ↔ Degrees: rad = deg × π/180, deg = rad × 180/π
Inverse trigonometric functions are the inverses of sine, cosine, and tangent. Note that arcsin and arccos have domain [-1,1], while arctan has domain all real numbers.
⚠Important: arcsin and arccos inputs must be within [-1,1]. Values outside this range cannot be computed.
What Are Inverse Trigonometric Functions?
Inverse trigonometric functions undo the trigonometric functions. If sin θ = x, then arcsin x = θ. A restricted range is necessary because a single sine value corresponds to infinitely many angles (due to periodicity). The principal values are used as the definition of the inverse functions.
Why Restrict the Range?
sin x = 0.5 has infinite solutions: 30°, 150°, 390°, … arcsin returns the principal value within the restricted range [-90°,90°], i.e., 30°.
arcsin + arccos Identity
arcsin x + arccos x = π/2 (90°). This follows from the identity sin θ = cos(90°-θ).
What are the domains and ranges of arcsin, arccos, arctan?▼
arcsin(x): domain [-1,1], range [-π/2, π/2] (-90° to 90°). arccos(x): domain [-1,1], range [0, π] (0° to 180°). arctan(x): domain (-∞,∞), range (-π/2, π/2) (-90° to 90°).
Because arcsin has a principal range of [-90°, 90°]. sin(150°) = 0.5, but 150° is outside this range. The principal value arcsin(0.5) = 30°. Use the relation sin(180°-θ)=sinθ to obtain 150°.
What is the relationship between inverse trig and regular trig functions?▼
Inverse trig functions are the inverses: sin(arcsin x) = x (within domain), arcsin(sin x) = x (within range). Note arcsin(sin x) ≠ x for all x; equality holds only when x is in [-π/2, π/2].
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