All subjects · Mathematics · WMA13 (P3)

Inverse trigonometric functions

arcsin, arccos and arctan are the inverse functions of sin, cos and tan. Because those are many-one, their domains are restricted first.

Restricted domains

These ranges give the principal value your calculator shows.

Graphs

Each graph is the reflection in y = x of the restricted graph of sin, cos or tan. y = arctan x has horizontal asymptotes y = ±π/2.

Using them

arcsin(1/2) = π/6, because sin(π/6) = 1/2 and π/6 is in the range. To find sin(arccos(3/5)), draw a right-angled triangle with cos θ = 3/5: the opposite side is 4, so sin θ = 4/5.

Worked example

Find the exact value of arccos(-1/2).

  1. cos(π/3) = 1/2.
  2. cos is negative in the second quadrant, so use π - π/3.
  3. 2π/3 lies in the range 0 to π.

Answer: 2π/3

Key idea: arcsin and arctan give values from -π/2 to π/2; arccos gives values from 0 to π. Each graph reflects the restricted trig graph in y = x.

Check yourself

  1. Find the exact value of arcsin(-1/2).
    • 7π/6
    • 11π/6
    • π/6
    • -π/6
    Show answer

    -π/6. sin(-π/6) = -1/2 and -π/6 lies between -π/2 and π/2.

  2. Find arctan(1) in radians, to 3 decimal places.
    Show answer

    0.785. arctan(1) = π/4 = 0.785.

  3. What is the range of y = arccos x?
    • -π/2 ≤ y ≤ π/2
    • -1 ≤ y ≤ 1
    • 0 ≤ y ≤ 2π
    • 0 ≤ y ≤ π
    Show answer

    0 ≤ y ≤ π. cos is one-one on 0 to π, so that is the restricted domain of cos and the range of arccos.

  4. Why must the domain of sin x be restricted before defining arcsin x?
    • sin x is many-one, so it must be made one-one
    • sin x is not continuous
    • sin x has asymptotes
    • sin x can be negative
    Show answer

    sin x is many-one, so it must be made one-one. Only one-one functions have inverses.

  5. Find the exact value of sin(arccos(3/5)). Give your answer as a fraction in its simplest form.
    Show answer

    4/5. Let θ = arccos(3/5). A 3, 4, 5 triangle gives sin θ = 4/5, positive because θ is between 0 and π.

  6. What is the domain of y = arcsin x?
    • -1 ≤ x ≤ 1
    • -π/2 ≤ x ≤ π/2
    • x ∈ ℝ
    • 0 ≤ x ≤ π
    Show answer

    -1 ≤ x ≤ 1. The inputs of arcsin are the outputs of sin, which lie between -1 and 1.

  7. Solve arctan(2x - 1) = π/4.
    Show answer

    1. 2x - 1 = tan(π/4) = 1, so x = 1.

  8. Solve arccos x = 1, where the angle is in radians. Give x to 3 decimal places.
    Show answer

    0.540. x = cos 1 = 0.540 (calculator in radians).

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Last updated 7 November 2026. Spotted a mistake? Tell us.