Rule: kisi function ko invert karne ke liye usse chosen piece par one-to-one (bijective) hona zaroori hai. Isliye wo sabse simple continuous interval chunte hain jo (a) har output value ek baar hit kare aur (b) x=0 ko achhi tarah include kare.
Sine[−π/2,π/2] par increase karta hai aur wahan saare outputs [−1,1] exactly ek baar cover karta hai. ✅ Ise chunte hain.
Cosine[−π/2,π/2] par one-to-one nahi hai (cos(−a)=cosa). [−1,1] ko ek baar cover karne wala agla sabse simple monotone interval [0,π] hai (decreasing). ✅ Ise chunte hain.
Tangent(−π/2,π/2) par increase karta hai aur wahan −∞ se +∞ tak sweep karta hai. Endpoints asymptotes hain → open interval.
Why must we restrict sine's domain to define arcsin?
Sine one-to-one nahi hai; [−π/2,π/2] tak restrict karne se har ratio ek unique angle se map hoti hai.
arcsinx+arccosx=?
π/2 for x∈[−1,1]
arccos(−x) in terms of arccosx
π−arccosx
cos(arcsinx)=?
1−x2
arcsin(sin(3π/4))=?
π/4 (snap into range)
Why is an inverse graph a reflection across y=x?
Kyunki (a,b)∈f⟺(b,a)∈f−1.
Is arcsin(sinx)=x always true?
Nahi — sirf x∈[−π/2,π/2] ke liye.
Recall Feynman: 12-saal ke bachche ko samjhao
Ek machine angle ko "slant number" mein badal deti hai. Kabhi kabhi tumhe slant number pata hoti hai aur tum jaanna chahte ho ki kaun sa angle usne banaya. Yahi inverse machine hai. Problem ye hai: kai angles ek hi slant number banate hain (jaise clock 30° aur 150° par sine ke liye ek jaisi lean karti hai). Isliye hum inverse machine ko bolte hain: "sirf ek choti safe zone ke angles mein se jawab dena." Wahi zone range hai. Ab har jawab unique hai aur kabhi confusing nahi hoga.
