InterviewSolution
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InterviewSolution
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i) Show that `sin^(-1){sin((3pi)/(4))} ne (3pi)/4` and find its value. ii) Show that `cos^(-1){cos(-pi/3)} ne -pi/3` and find its value. iii) Show that `tan^(-1){tan((5pi)/6)}ne (5pi)/6` and find its value. |
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Answer» i) We know that the principal-value branch of `sin^(-1)` is `[-pi/2,pi/2]`. `therefore sin^(-1){sin(3pi)/4} ne (3pi)/4`. Now, `sin^(-1){sin(3pi)/4)=sin^(-1){sin(pi-pi/4)}=sin^(-1){sinpi/4}[therefore sin(pi-pi/4)=sinpi/4]` `=pi/4 {therefore pi/4 in [-pi/2, pi/2]}`. `therefore sin^(-1){sin (3pi)/4}=pi/4`. ii) We know that the principal-value branch of `cos^(-1){cos(-pi/3)} ne -pi/3`. Now, `cos^(-1){cos(-pi/3)}=cos^(-1){cospi/3} [therefore cos(-theta)=costheta]=pi/3 [therefore pi/3 in [0,pi]]`. `therefore cos^(-1){cos(-pi/3)}=pi/3`. iii) We know that the principal-value branch of `tan^(-1)` is `(-pi/2,pi/2)`. `therefore tan^(-1){tan(5pi)/6} ne (5pi)/6`. Now, `tan^(-1)(tan(5pi)/6}=tan^(-1){tan(pi-pi/6)}=tan^(-1){tan(-pi/6)} [therefore tan(pi-theta)=tan(-theta)]`. `=-pi/6 [therefore -pi/6 in (-pi/2, pi/2)]` `therefore tan^(-1){tan(5pi)/6}=-pi/6`. |
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