Choose a principal inverse branch

Try

You know sin θ=1/2. Both π/6 and 5π/6 fit, and adding full turns creates infinitely many more angles. A function named arcsin must return one output, so it selects a particular branch. Which branch can be used without losing any sine value between −1 and 1?

Predict

Predict whether arcsin(sin(5π/6)) returns the original angle. It cannot if arcsine is restricted to angles between −π/2 and π/2. Distinguish reversing a restricted function from solving an unrestricted trigonometric equation.

Experiment

Create three sketches: sine restricted to [−π/2,π/2], cosine restricted to [0,π], and tangent restricted to (−π/2,π/2). Each restricted graph passes a horizontal-line uniqueness test.

Inverse	Input domain	Principal output range
arcsin u	−1≤u≤1	−π/2≤θ≤π/2
arccos u	−1≤u≤1	0≤θ≤π
arctan u	All real u	−π/2<θ<π/2

Check endpoint inclusion. Arcsine and arccosine can return their finite endpoint angles. Arctangent cannot return ±π/2 because tangent is undefined there.

Understand

An inverse function reverses the selected one-to-one branch. Thus sin(arcsin u)=u for u in [−1,1]. In the opposite order, arcsin(sin θ)=θ only when θ lies within the chosen principal range. For θ=5π/6, the sine is 1/2 and the returned angle is π/6.

For cos(arcsin(4/5)), introduce θ=arcsin(4/5). Its principal range has nonnegative cosine. The identity cos²θ=1−sin²θ gives cos θ=3/5, choosing the positive square root. Without a range or quadrant, knowing a sine alone would not determine the sign of cosine.

Arctangent of a quotient can also lose quadrant information. The ratios 1/1 and (−1)/(−1) agree, but their points lie in opposite quadrants. A Cartesian-to-angle conversion must inspect both coordinate signs or use an appropriate two-argument angle procedure. A single inverse function cannot infer information absent from its input.

The notation sin⁻¹ often denotes arcsine, not 1/sin. Use the names arcsin and cosecant when ambiguity matters. A reciprocal returns a number formed by division; an inverse trigonometric function returns an angle with a specified branch and unit.

Apply

In practice, give the rational coefficient k when an angle is requested as kπ. For the written general-solution task, include the other symmetry branch and all integer periods. A principal calculator output is a useful starting point, not a complete answer to an all-real-angle equation.

Continue

Identities connect trigonometric expressions on shared domains. The next lesson uses algebraic justification and counterexamples rather than relying on numerical agreement at selected angles.