Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: Inverse trigonometric functions return the one angle in a restricted interval that has a given sine, cosine or tangent: arcsin gives angles in [−π/2, π/2], arccos in [0, π], and arctan in (−π/2, π/2). Solving trig equations means finding all angles in one period and adding multiples of the period.
Standards: F-TF.B.6, F-TF.B.7 · Level: Grade 12 Precalculus
Why do inverse trigonometric functions need restricted domains?
Sine, cosine and tangent repeat, so they fail the horizontal line test. To invert them we keep one piece on which each is one-to-one. The inverse then returns exactly one angle.
| Inverse | Input (domain) | Output (range) | Based on |
|---|---|---|---|
| arcsin x | [−1, 1] | [−π/2, π/2] | sin on [−π/2, π/2] |
| arccos x | [−1, 1] | [0, π] | cos on [0, π] |
| arctan x | all real numbers | (−π/2, π/2) | tan on (−π/2, π/2) |
The notation sin−1 x means arcsin x, not 1/sin x. The value arcsin(1/2) = π/6 asks: which angle in [−π/2, π/2] has sine 1/2?
Assuming arcsin(sin x) = x for every x. That is true only when x is in the range [−π/2, π/2]. For example arcsin(sin 5π/6) = arcsin(1/2) = π/6, not 5π/6.
How do you solve a trigonometric equation?
- Isolate the trigonometric expression (factor or use a quadratic form if needed).
- Use the inverse function and symmetry to list all solutions in one period: [0, 2π) for sine and cosine, [0, π) for tangent.
- For all real solutions add 2πk (sine, cosine) or πk (tangent), where k is an integer.
- For a restricted interval, keep only the values that fit.
Solve 2 sin x = √3. Then sin x = √3/2, with reference angle π/3 and sine positive in Quadrants I and II. On [0, 2π): x = π/3 and 2π/3. In general: x = π/3 + 2πk or 2π/3 + 2πk.
When the number is not special, use technology in radian mode. If cos x = 0.3, the calculator gives arccos 0.3 ≈ 1.2661, and symmetry gives the second solution 2π − 1.2661 ≈ 5.0171. In a word problem, check that each solution fits the context before reporting it.
Never divide both sides by a trigonometric factor such as cos x. Move everything to one side and factor, so no solutions disappear.
- arcsin has range [−π/2, π/2]; arccos has range [0, π]; arctan has range (−π/2, π/2).
- sin⁻¹ x means arcsin x, not 1/sin x.
- arcsin(sin x) = x only when x is in [−π/2, π/2].
- General solutions add 2πk for sine and cosine and πk for tangent.
- Factor trigonometric equations instead of dividing by a trig factor.
Inverse Trigonometric Functions and Trig Equations: frequently asked questions
Why do inverse trigonometric functions need restricted domains?
Sine, cosine and tangent are periodic, so many inputs share the same output and the functions are not one-to-one. Restricting to one piece, such as sine on [−π/2, π/2], makes an inverse function possible.
Is sin⁻¹(x) the same as 1/sin(x)?
No. The notation sin⁻¹(x) denotes the inverse function arcsin(x), which returns an angle. The reciprocal 1/sin(x) is the cosecant. Check the context to avoid mixing them up.
Why is arcsin(sin x) not always equal to x?
The output of arcsin must lie in [−π/2, π/2]. If x is outside that interval, arcsin returns the angle in that interval with the same sine, so arcsin(sin(5π/6)) = π/6.
How do you find all solutions of a trigonometric equation?
Solve for all angles in one period using the inverse function and symmetry, then add multiples of the period: 2πk for sine and cosine, πk for tangent. Restrict the list if an interval is given.
How do you solve a trigonometric equation with a calculator?
Set the calculator to radian mode, apply the inverse function to get one solution, and use symmetry to get the others in the period. Round only at the end and check that each answer fits the situation.
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