Answer to Question #156106 in Trigonometry for .

Question #156106

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


1
Expert's answer
2021-01-19T03:45:35-0500

a) cotα+1sinβ=1sin2α1+11cos2β=\cot{\alpha} +\frac{1}{\sin{\beta}}=\sqrt{\frac{1}{\sin^2{\alpha}}-1}+\frac{1}{\sqrt{1-\cos^2{\beta}}}= 2591+111625=43+53=3\sqrt{\frac{25}{9}-1}+\frac{1}{\sqrt{1-\frac{16}{25}}}=\frac{4}{3}+\frac{5}{3}=3

b)sin2αtan2β=2sinαcosα×2sinβcosβcos2βsin2β=\sin{2\alpha}\tan{2\beta}=2\sin{\alpha}\cos{\alpha}\times \frac{2\sin{\beta}\cos{\beta}}{\cos^2{\beta}-\sin^2{\beta}}= 2sinα1sin2α×2cosβ1cos2βcos2β1+cos2β=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}}= 651925×851162532251=2425×(247)=576175\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}


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