Mathfolis

Inverse Trigonometric Functions

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

Inverse trig functions reverse sin, cos, tan — but only on restricted domains, since the originals are not one-to-one. The principal ranges are: arcsin⁡:[−π/2,π/2]\arcsin: [-\pi/2, \pi/2], arccos⁡:[0,π]\arccos: [0, \pi], arctan⁡:(−π/2,π/2)\arctan: (-\pi/2, \pi/2). Compositions like sin⁡(arcsin⁡x)\sin(\arcsin x) always simplify to xx within the domain, but the reverse arcsin⁡(sin⁡θ)=θ\arcsin(\sin\theta) = \theta only when θ\theta is in the principal range.

Principal ranges

arcsin
[−1,1]→[−π/2,π/2][-1, 1] \to [-\pi/2, \pi/2]
arccos
[−1,1]→[0,π][-1, 1] \to [0, \pi]
arctan
R→(−π/2,π/2)\mathbb{R} \to (-\pi/2, \pi/2)
Caution: arcsin⁡(sin⁡θ)\arcsin(\sin\theta) does NOT always equal θ\theta. If θ=2π/3\theta = 2\pi/3, then sin⁡θ=3/2\sin\theta = \sqrt 3/2, and arcsin⁡(3/2)=π/3≠2π/3\arcsin(\sqrt 3/2) = \pi/3 \neq 2\pi/3.
Type 1

Evaluate arcsin, arccos, arctan

Recall the special-angle value whose sin (or cos, tan) matches the input, picking the one in the principal range.

Example 1
What is arctan⁡(1)\arctan(1)? (A) π/6\pi/6 (B) π/4\pi/4 (C) π/3\pi/3 (D) π/2\pi/2

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 2

Domain and range

arcsin and arccos accept inputs in [−1,1][-1, 1]; arctan accepts all reals. Their outputs lie in the principal ranges.

Example 2
What is the range of arccos⁡\arccos? (A) [−π/2,π/2][-\pi/2, \pi/2] (B) [0,π][0, \pi] (C) (−π/2,π/2)(-\pi/2, \pi/2) (D) [−π,π][-\pi, \pi]

Practice more of this type— AI-generated · always-new problems

Generate Problems →
Type 3

Composition with original trig

When evaluating sin(arccos x) or cos(arcsin x), use a right triangle (or the Pythagorean identity) to find the missing side.

Example 3
Evaluate cos⁡(arcsin⁡(3/5))\cos(\arcsin(3/5)). (A) 3/53/5 (B) 4/54/5 (C) 5/35/3 (D) 5/45/4

Practice more of this type— AI-generated · always-new problems

Generate Problems →