AP PRECALCULUS — UNIT 3B · TRIGONOMETRIC & POLAR FUNCTIONS
3.9 — Inverse Trigonometric Functions
Notes — Undoing Sine, Cosine, and Tangent
💡 Learning Objectives (3.9.A)
By the end of this lesson you will be able to:
- Define arcsin, arccos, and arctan as inverse functions on restricted domains
- Identify the domain and range of each inverse trig function
- Evaluate inverse trig expressions exactly for common values
- Compose trig and inverse trig functions correctly, using restricted ranges
1. Why Restrict the Domain?
Sine, cosine, and tangent are NOT one-to-one over their full domains — many inputs give the same output. To define an inverse function, we restrict each one to a domain where it IS one-to-one, then invert.
The standard restrictions are chosen to (a) make the function one-to-one and (b) include the largest possible range of outputs that contains the easy-to-state Quadrant I.
2. The Three Inverse Trig Functions
💡 Domain and Range Summary
- y = arcsin(x) (also sin⁻¹ x): Domain [−1, 1], Range [−π/2, π/2]
- y = arccos(x) (also cos⁻¹ x): Domain [−1, 1], Range [0, π]
- y = arctan(x) (also tan⁻¹ x): Domain (−∞, ∞), Range (−π/2, π/2)
Notice that arcsin and arctan share the range (−π/2, π/2) — Quadrants I and IV on the unit circle (and the boundary). arccos lives in Quadrants I and II — the range [0, π].
⚠️ Common mistake
sin⁻¹(x) does NOT mean 1/sin(x). The −1 superscript indicates ‘inverse function,’ not a reciprocal. The reciprocal of sin(x) is csc(x), which we'll meet in 3.11.
3. Evaluating Inverse Trig Functions
To evaluate arcsin(c), arccos(c), or arctan(c), ask: ‘What angle in the inverse function's RANGE has the requested trig value c?’
📘 Example — Exact values
- arcsin(1/2) = π/6 (because sin(π/6) = 1/2 and π/6 ∈ [−π/2, π/2])
- arcsin(−√2/2) = −π/4 (in [−π/2, π/2], not 5π/4 even though sin(5π/4) is also −√2/2)
- arccos(−1/2) = 2π/3 (because cos(2π/3) = −1/2 and 2π/3 ∈ [0, π])
- arctan(√3) = π/3
- arctan(−1) = −π/4
4. Composing Trig and Inverse Trig
There are two natural compositions: ‘sin of arcsin’ and ‘arcsin of sin.’ The first is always straightforward; the second has a catch.
💡 Composition Rules
- sin(arcsin(x)) = x for every x in [−1, 1]
- arcsin(sin(θ)) = θ ONLY when θ ∈ [−π/2, π/2]
- cos(arccos(x)) = x for every x in [−1, 1]
- arccos(cos(θ)) = θ ONLY when θ ∈ [0, π]
- tan(arctan(x)) = x for every real x
- arctan(tan(θ)) = θ ONLY when θ ∈ (−π/2, π/2)
📘 Example — When the inner θ is outside the range
Compute arcsin(sin(5π/6)).
- First find sin(5π/6) = 1/2
- Then arcsin(1/2) = π/6 (NOT 5π/6, because 5π/6 ∉ [−π/2, π/2])
- Final answer: π/6
5. Mixed Compositions
Expressions like cos(arcsin(3/5)) or tan(arccos(−1/2)) come up on the AP exam. Use a right-triangle interpretation: let θ = arcsin(3/5), so sin(θ) = 3/5 with θ in [−π/2, π/2]. In that range, cos(θ) ≥ 0, so by Pythagoras cos(θ) = 4/5. Therefore cos(arcsin(3/5)) = 4/5.
📘 Example — A trickier composition
Find tan(arccos(−1/2)).
- Let θ = arccos(−1/2), so cos(θ) = −1/2 and θ ∈ [0, π].
- That puts θ = 2π/3 (Quadrant II).
- sin(2π/3) = √3/2
- tan(2π/3) = sin/cos = (√3/2) / (−1/2) = −√3
6. Graphs of Inverse Trig Functions
Each inverse function's graph is the reflection of the corresponding trig function's restricted graph across y = x:
- y = arcsin(x): increasing, S-shape, passes through (0, 0), endpoints (−1, −π/2) and (1, π/2)
- y = arccos(x): decreasing, passes through (0, π/2), endpoints (−1, π) and (1, 0)
- y = arctan(x): increasing, S-shape, horizontal asymptotes y = ±π/2, passes through (0, 0)
7. Summary
- Inverses require restricted domains; the chosen ranges are the standard ‘principal values’
- arcsin: [−π/2, π/2]; arccos: [0, π]; arctan: (−π/2, π/2)
- Composition is symmetric only when the inner output stays within the range
- Use right-triangle reasoning (or the unit circle) for mixed compositions
- sin⁻¹ is NOT 1/sin — it is the inverse function