Learning goal and prerequisites
Distinguish inverse evaluation from unrestricted equation solving. Before starting, review trig equations.
Method and conditions
Arcsin returns angles in [−π/2,π/2]; arccos returns [0,π]; arctan returns (−π/2,π/2). Reduce an inner angle using unit-circle symmetry, then choose the representative in the inverse function’s range.
Worked example: show the decisions
arcsin(sin150°)=arcsin(1/2)=30°. The sine equation has other answers, but the inverse function has a single output. In contrast arccos(cos240°)=arccos(−1/2)=120°, because arccos uses the upper semicircle.
Contrast: why a plausible shortcut fails
The composition inverse(trig(θ)) need not equal θ outside its principal interval. Also sin⁻¹(x) denotes inverse sine in this context; it is not the reciprocal 1/sin(x).
Transfer: try before revealing the answer
Evaluate arctan(tan135°) and arcsin(sin210°), in degrees.
Transfer solution and check
The first is −45°, since tangent has period 180° and −45° lies inside the arctan interval. The second is −30°, because sin210°=−1/2 and arcsin selects −30°.
Evaluate arcsin(sin(120°)) in degrees using the principal inverse sine.
- Choose the angle with the same sine in [−90°,90°].
- Using these quantities: 180-120 = 60.
- Check the original domain, requested quantity and units; do not substitute a related parameter.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus Course at a Glance