Saved Bookmarks
| 1. |
If X and Y are acute angels such that sinX=1/√5 and sinY =1/√10 prove that (x+y)=π/4. |
| Answer» Ans. Given : {tex}sin X = {1\\over \\sqrt 5}{/tex},\xa0{tex}sin Y = {1\\over \\sqrt {10}}{/tex}X =\xa0{tex}sin^{-1}({1\\over \\sqrt 5}){/tex}Y =\xa0{tex}sin^{-1}({1\\over \\sqrt {10}}){/tex}=> X+Y =\xa0{tex}sin^{-1}({1\\over \\sqrt 5}) + sin^{-1}({1\\over \\sqrt {10}}){/tex}{tex}[using \\ \\ sin^{-1} a + sin^{-1} b = sin^{-1}(a\\sqrt{1-b^2}+b\\sqrt{1-a^2})]{/tex}{tex}=> X+Y = sin^{-1}\\left ( {1\\over \\sqrt 5} \\sqrt {1- {1\\over 10}} + {1\\over \\sqrt {10}} \\sqrt {1- {1\\over 5}}\\right ){/tex}\xa0{tex}=> X+Y = sin^{-1}\\left ( {3\\over \\sqrt {50}} + {2\\over \\sqrt {50}} \\right ){/tex}{tex}=> X+Y = sin^{-1}\\left ( {5\\over 5\\sqrt 2} \\right ){/tex}{tex}=> X+Y = sin^{-1}\\left ( {1\\over \\sqrt 2} \\right ){/tex}{tex}=> X+Y = sin^{-1}\\left ( sin {\\pi \\over 4} \\right ){/tex}{tex}=> X+Y = {\\pi \\over 4} {/tex}Hence Proved | |