Inverse Trigonometric Functions

Objective: Explain principal arcsine and arccosine, apply the rule, and justify relevant restrictions.

Sine and cosine are not one-to-one on all real angles. Define arcsin by restricting sine to [−π/2,π/2], and arccos by restricting cosine to [0,π]. Both inverse functions have domain [−1,1]. These principal outputs select one representative angle, while a trigonometric equation can have other solutions outside the restricted range.

Objective: Explain arctangent and compositions, apply the rule, and justify relevant restrictions.

Restrict tangent to (−π/2,π/2) to define arctan with domain all real numbers. Then tan(arctan u)=u for every real u. The reverse composition arctan(tan θ)=θ only for θ in the restricted interval. For other angles, the principal result is coterminal modulo π within that interval. Track whether a composition returns a ratio or an angle.