Question

given sin α = 3/5 and cos β = -4/5 , α is an acute angle and β is an obtuse angle find ;

a) cot α + cosec β

b) sin 2α tan 2β

Answer 1

a) $\cot{\alpha} +\frac{1}{\sin{\beta}}=\sqrt{\frac{1}{\sin^2{\alpha}}-1}+\frac{1}{\sqrt{1-\cos^2{\beta}}}=$ $\sqrt{\frac{25}{9}-1}+\frac{1}{\sqrt{1-\frac{16}{25}}}=\frac{4}{3}+\frac{5}{3}=3$

b)$\sin{2\alpha}\tan{2\beta}=2\sin{\alpha}\cos{\alpha}\times \frac{2\sin{\beta}\cos{\beta}}{\cos^2{\beta}-\sin^2{\beta}}=$ $2\sin{\alpha}\sqrt{1-\sin^2{\alpha}}\times \frac{2\cos{\beta}\sqrt{1-\cos^2{\beta}}}{\cos^2{\beta}-1+\cos^2{\beta}}=$ $\frac{6}{5}\sqrt{1-\frac{9}{25}}\times \frac{-\frac{8}{5}\sqrt{1-\frac{16}{25}}}{\frac{32}{25}-1}=\frac{24}{25}\times(-\frac{24}{7})=-\frac{576}{175}$