1.

Find the value of sin^-1⁡(\(\frac{5}{13}\))+cos^-1⁡(\(\frac{3}{5}\)).(a) sin^-1⁡(\(\frac{63}{65}\))(b) sin^-1⁡1(c) 0(d) sin^-1⁡(\(\frac{64}{65}\))This question was addressed to me in semester exam.My query is from Properties of Inverse Trigonometric Functions topic in division Inverse Trigonometric Functions of Mathematics – Class 12

Answer»

The CORRECT OPTION is (a) sin^-1⁡(\(\frac{63}{65}\))

Best explanation: From ∆ABC, we get

cos^-1⁡\((\frac{3}{5})\)=sin^-1⁡\((\frac{4}{5})\)

∴sin^-1⁡(\(\frac{5}{13}\))+cos^-1⁡(\(\frac{3}{5}\))=sin^-1⁡(\(\frac{5}{13}\))+sin^-1⁡(\(\frac{4}{5}\))

=sin^-1⁡\((\frac{5}{13}\SQRT{1-(\frac{4}{5})^2}+\frac{4}{5}\sqrt{1-(\frac{5}{13})^2})\)

=\(sin^{-1}(\frac{5}{13}×\frac{3}{5}+\frac{4}{5}×\frac{12}{13})=sin^{-1}(\frac{15+48}{65})=sin^{-1}(\frac{63}{65})\).


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