Inverse trigonometric functions
Sine, cosine and tangent are not injective, so their inverses need restricted domains. The standard ranges are arcsin: [−π/2, π/2], arccos: [0, π], and arctan: (−π/2, π/2). This is why arcsin(sin(3π/4)) is π/4 rather than 3π/4 — the inverse can only return values inside its range.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Remember the three rangesarcsin and arctan return values in the right half of the circle (Q1 and Q4); arccos returns the top half (Q1 and Q2).
- Evaluate compositions carefullyarcsin(sin x) = x only when x is already in [−π/2, π/2]. Otherwise find the angle in range with the same sine.
- Use a triangle for mixed compositionsFor cos(arcsin x), draw a right triangle with opposite x and hypotenuse 1, giving adjacent √(1−x²).
- Check the domainarcsin and arccos accept only inputs in [−1, 1]; arctan accepts everything.
Worked example
Evaluate arcsin(sin(3π/4)) and cos(arcsin(3/5)).
- sin(3π/4) = √2/2. arcsin must return a value in [−π/2, π/2].
- The angle in that range with sine √2/2 is π/4, so arcsin(sin(3π/4)) = π/4.
- For the second: let θ = arcsin(3/5), so sin θ = 3/5 with θ in Q1.
- Then cos θ = √(1 − 9/25) = 4/5, positive since θ is in Q1.
Answer. arcsin(sin(3π/4)) = π/4 and cos(arcsin(3/5)) = 4/5.
Where marks get dropped
These are the specific errors that cost credit on inverse trigonometric functions questions — QED's rubric penalises each of them separately.
- Assuming arcsin(sin x) = x for every x. It only holds inside the restricted range.
- Writing sin⁻¹x as 1/sin x. The superscript −1 denotes the inverse FUNCTION here, not a reciprocal — that is csc x.
- Ignoring the range when solving equations. The inverse gives one solution; the others come from symmetry.
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Inverse trigonometric functions — frequently asked questions
Why restrict the range at all?
Because sine is not injective — infinitely many angles share each value. Restricting to [−π/2, π/2] makes it injective so an inverse function exists.
Is sin⁻¹x the same as csc x?
No. sin⁻¹x is arcsin, the inverse function; csc x = 1/sin x is the reciprocal. The clash of notation is unfortunate but standard.
What is the derivative of arctan?
1/(1 + x²), which is why arctan appears throughout integration — ∫dx/(1+x²) = arctan x + C.
The rest of Trigonometry
Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Right-triangle ratios: sin, cos & tan
- 2The unit circle & exact values
- 3Radians, degrees & arc length
- 4Graphs of sin, cos & tan
- 5Pythagorean & reciprocal identities
- 6Angle-sum, difference & double-angle formulas
- 7Solving trigonometric equations
- 8The sine rule & the cosine rule
- 9Triangle area, sectors & segments
- 10Inverse trigonometric functions
- 11Proving trigonometric identities
- 12Polar coordinates & converting to Cartesian
- 13Modelling periodic phenomena with sinusoids
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