​ (i) Write the principal value of sin^−1(1/2) (ii) Show that sin ^-1 (3/5) - sin^-1(8/17) = cos^-1 (84/85)​

Question

(i) Write the principal value of sin⁻¹(\(\frac{1}{2}\))

(ii) Show that

sin ^-1 (\(\frac{3}{5}\)) - sin^-1 \(\frac{8}{17}\) = cos^-1 \(\frac{84}{85}\)

Answer 1

(i) sin^-1 \((\frac{1}{2})\) = \(\frac{π}{6}\)

(ii) Let sin^-1 (\(\frac{3}{5}\)) = x and sin^-1 (\(\frac{8}{17}\)) = y

sin x = \(\frac{3}{5}\) ⇒ cos x = \(\sqrt{1-sin^2x}\)

= \(\sqrt{1- (\frac {3}{5})^2}\) = \(\frac{4}{5}\) ; sin y = \(\frac{8}{17}\)

cos (x - y) = cos x cos y + sin x sin y

=