Question

sin^-1⁡x in terms of cos^-1⁡is ____________

(a) cos^-1⁡\(\sqrt{1+x^2}\)

(b) cos^-1⁡\(\sqrt{1-x^2}\)

(c) cos^-1⁡x

(d) cos^-1\(\frac{⁡1}{x}\)

The question was asked in a national level competition.

The question is from Properties of Inverse Trigonometric Functions in section Inverse Trigonometric Functions of Mathematics – Class 12

Answer 1

Right choice is (b) cos^-1⁡\(\sqrt{1-x^2}\)

The explanation is: Let sin^-1⁡x=y

⇒x=sin⁡y

⇒\(x=\sqrt{1-cos^2⁡y}\)

⇒\(x^2=1-cos^2⁡y\)

⇒\(cos^2⁡y=1-x^2\)

∴y=cos^-1⁡\(\sqrt{1-x^2}\)=sin^-1⁡x.